Algebra II
Probability Notes
Probability measures how likely it is for an event to occur.
When you gather data from observations, you can calculate an experimental probability. Each observation is
called an experiment or trial.
Experimental Probability of an Event =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
Simulation = models an event
Sample space = set of all possible outcomes
Theoretical Probability = If a sample space has n equally likely outcomes and an event A occurs in m of those
events, then the theoretical probability if event A
𝑚
is 𝑃(𝐴) = 𝑛 .
Practice Problems
1. You select a number at random from the sample space {1, 2, 3, 4, 5}. Find each theoretical probability.
1
 0.20
5
a. P(the number is 2)
c. P(the number is prime)
{‘1’ is NOT prime}
e. P( 1 or 3)
P (1)  P(3)
3
 0.60
5
2
 0.40
5
2
 0.40
5
b.
P(the number is even)
d.
P(the number is less than 5)
f.
P(the number is not 2)
P(not 2)  1  P(2)
1 4
1 
5 5
" winners " 2

total
5
1 1

5 5
1
4
 0.80
5
4
 0.80
5
One of these names is to be drawn from a hat. Determine each probability below: (10 total)
Mary
Jenny
2. P(3-letter name) =
Bob
Marilyn
2
1
or
10 5
Bill
Jack
Jerry
Tina
Connie
(What is the probability of drawing a 3-letter name?)
2 1
4 2
  0.20
  0.40
10 5
3. P(4-letter name) = _____________
4. P(name starting with B) = ____________
10 5
1
(Mary, Bill, Jack, Tina)
 0.10
5. P(name starting with T) = __________
10
0
(tina)
0
7. P(name starting with S) = __________
10
1
(Bob, Bill)
 0.10
6. P(7-letter name) = ______________
10
3
(Marilyn)
 0.30
8. P(name ending with Y) = _____________
10
(Mary, Jenny, Jerry)
One of these cards will be drawn without looking. (12 total)
4
10
7
J
S
9
10
2
M
5
4
J
number of twos
12 total number of cards
9. P(2) = 1
1
10. P(5) = ________
12
2 1

13. P(4) = ________
12 6
8 2

12 3
12. P(a number) = _________
(10,4,7,9,10,2,5,4) 4  1
15. P(a letter) = __________
12 3
2 1

11. P(J) = _________
12 6
0
14. P(T) = _________
12
One card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing…
4
1
12 3
(4J, 4Q,4K)


52
13
52 13
16. P(ace) = ________
17. P(face card) = _________
2
1

18. P(a red 10) = ________
52 26
1  P(diamond ) 
3
13
1 3
19. P(NOT a diamond) = ______________
4
1
 1 
52
4 4
A spinner, numbered 1–8, is spun once. What is the probability of spinning… (8 total)
2
1
3
8
4
7
5
4 1

8 2
19. an EVEN number? _________
2 1

8 4
20. a multiple of 3? ___________
4 1

21. a PRIME number? _________
8 2
0
8
22. 9? ____________
6
2
Joe
Law of Large Numbers:
Investigation:
1) Toss a thumbtack 20 times on a table (not on a piece of paper). Make a tally chart to record the number of
times, the pointy side is down.
2) Record your total number in the first row in the chart below. Ask another student for their total. Find the
average of your two tosses. Ask another student for their total, and find the average of the three totals you have.
Continue asking other students, and finding the running average.
Name:
# down per person.
Total down.
Total # tosses
3) Graph the points using the last two columns as coordinates.
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
20
40
60
80 100 120 140 160 180 200 220 240 260
3
Running Average
4) Class observations about the running average?
The averages should converge to the relative frequency/probability.
5) Law of Large numbers:
The long run relative frequency of repeated independent events will settle
down to the true probability as the number of trials increases.
6) Law of Averages:
a) Boy, I was positive I was going to get an A on the last test. I have gotten so many C’s, I figure I was
due to get a good grade soon.
b) I wasn’t scared that the Bulls would lose their game to the Oklahoma City Thunder because the
Thunder have won so many in a row, they were due for a loss. Luckily, they were playing the
Bulls. Comment on the logic.
c) My brother decided to settle an argument by tossing a coin and calling it in the air. I called tails,
but lost. He offered double of nothing (which meant I had to take out the garbage for two
months, not one if I lost). I took the deal because I knew the next toss would be tails because the
last toss was heads and probability says that 50% of the time, it will be heads. Since it wasn’t
last time, it had to be this time.
Comment on the logic of the statements above.
This is referred to as the LAW of SMALL MINDS. Independent events will
somehow compensate and make the probability come true in the short run.
It assumes that some sort of compensatory mechanism in the universe exists to
balance out the results.
7. After an unusually dry autumn, a radio announcer is heard to say, “Watch out! We’ll pay for these sunny
days later on this winter.” Explain what he’s trying to say, and comment on the validity of his
reasoning.
The announcer is using the Law of Averages. The weather does not keep track of
what the weather was on previous days.
8. A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run.
When talking to reporters afterward, he says he was very confident that last time at bat because he knew
he was “due for a hit.” Comment on his reasoning.
There is no universal force balancing things out. If he bats a lot of times, his
average would become accurate. In the short run, there is no such thing as being
‘Due’ for a hit.
4
Mutually Exclusive:
9.
In a telephone survey of 150 households, 75 respondents answered “Yes” to a particular question, 50
answered “No,” and 25 were “Not sure.” Find each experimental probability.
Venn Diagram:
No
Yes
75
a. P(answer was “Yes”)
75 1
  0.50
150 2
d. P(answer was not “Yes”)
50
b. P(answer was “No”)
50 1
  0.33
150 3
e. P(answer was not “No”)
1- P(Yes)
1 – P(no)
Or
or
P(No) + P(Not Sure)
P(Yes) + P(not sure)
0.50
0.67
Not sure
25
c. P(answer was “Not sure”)
25 1
  0.17
150 6
f. P (answer Yes or No)
P(Yes) + P(No)
75 50

150 150
125 5
  0.83
150 6
Mutually Exclusive (also known disjoint) sets:
Events are disjoint or mutually exclusive if the two events do not have any
outcomes in common.
5
10. In a class of 19 students, 10 study Spanish, 7 study French, and 2 study both French and Spanish. One
student is picked at random. Find each probability.
Let’s make a Venn Diagram to represent the information:
French
Spanish
8
5
2
Neither
4
a. P(studying Spanish but not French) b. P(studying neither Spanish nor French)
8
 0.42
19
c. P(studying French)
4
 0.21
19
d. P(studying Spanish)
10
 0.53
19
7
 0.37
19
e. P(studying Spanish or French)
{using diagram and using formula}
8  2  5 15

 0.79
19
19
10 7 2 15
  
 0.79
19 19 19 19
11. Since there is some “overlap” of the data, we say the categories are:
NOT MUTUALLY EXCLUSIVE
12. Probability Formula:
P( A  B)  P( A)  P( B)  P( A  B)
{Did this formula work for #9F, which was another OR problem?}
Yes, but since there was no overlap, we just subtracted 0.
6
13. In Mrs. Esgar’s 3rd period Alg. II class of 25 students. 12 people from the class took part in the Fall play. 4
of the people in the class had to miss play practice to compete in a mathletes competition, while 10
people from the class did not miss play practice (because they were not in the play) for the mathletes
competition. In Mrs. Sager’s 3rd period Alg. II class of 20 students. 10 people from the class took part
in the fall play. 8 of the people in the class had to miss play practice to compete in a mathletes
competition, while 6 of the people from class did not miss play practice for the mathletes competition
(since they were a mathlete and not a fall play person).
a) Make a Venn diagram for Mrs. Esgar’s class that are in the fall play and in mathletes.
Esgar
Mathletes
Fall Play
8
10
4
Neither
3
i) Find the probability of being in Mrs. Esgar’s class and being in the fall play.
ii) Find the probability of being in Mrs. Esgar’s class and being a mathlete.
12
 0.48
25
14
 0.56
25
iii) Find the probability of being in Mrs. Esgar’s class and being in mathletes or the fall play.
12 14 4 22
8  4  10 22



OR

25 25 25 25
25
25
iv) Are Mathletes and Fall Play mutually exclusive sets? Explain.
No, the events have some overlap, so they are not mutually exclusive.
b) Make and label a Venn diagram, using the given data for Mrs. Esgar’s students and Mrs. Sager’s
students. (Don’t worry about the play or mathletes, just work with students)
Esgar
Sager
25
20
i) Are the sets mutually exclusive? How do you know?
Yes, there is no overlap of students, so the sets are mutually exclusive. A student
cannot be in Mrs. Esgar’s 3rd period class and also Mrs. Sager’s 3rd period class.
7
14. In Mrs. Esgar’s 3rd period Alg. II class of 25 students. 12 people from the class took part in the Fall play. 4
of the people in the class had to miss play practice to compete in a mathletes competition, while 10 people from
the class did not miss play practice for the mathletes competition. In Mrs. Sager’s 3rd period Alg. II class of 20
students. 10 people from the class took part in the fall play. 8 of the people in the class had to miss play
practice to compete in a mathletes competition, while 6 of the people from class did not miss play practice for
the mathletes completion.
a) Make a diagram to help organize ALL of the data for the two classes.
Mrs. Sagers 3rd period class
Mrs. Esgar 3rd period class
Mathletes
Fall Play
8
Mathletes
Fall Play
10
4
2
8
6
4
3
3rd period classes (45 students total)
Mathletes
Mrs. Esgar
3
6
10
4
8
8
2
Mrs. Sagers
4
Fall Play
b) P(selecting a mathlete) from the students in Ms. Sagers or Ms. Esgar’s 3rd period class.
P(Sager Mathlete ) + P(Esgar Mathlete)
14  14 28

 .62
20  25 45
c) P(selecting a mathlete or a Fall Play participant) from the students in Ms. Sagers or Ms.
Esgar’s 3rd period class.
2  8  6  8  4  10 38

 .84
20  25
45
4  3 38
1  P (neither )  1 

 .84
45
45
8
d) If want to select one student from Mrs. Sager’s class and one student from Mrs. Esgar’s class, would
our choice for Mrs. Esgar’s class be affected by the choice of Mrs. Sager’s class?
No. Since there is no overlap… the first choice will not affect the 2nd choice, they would be
independent choices.
e) When the first selection has NO effect on the probability of the next selection:
We say that the events are INDEPENDENT of each other.
If the 1st choice will in any way affect the choice or probability of the 2nd event,
then the events are NOT INDEPENDENT.
15. The M&M company has published the percentages of the 6 colors found in each bag of M&Ms. {I
am going to make some up right now, if you care, look them up} In a bag of M&M’s there should be
25% browns, 20% reds, 10% orange and 10% yellows, 15% blues and 2 % greens.
a) What is wrong with the data, and how did you know?
The theoretical percentages are supposed to add up to 100%. They only add up to
82%. There was a typo for the greens. It was supposed to be 20%.
b) If we randomly reach into a bag of M&M’s and select ONE, what is the probability that it is Blue?
0.15
c) If we randomly reach into a bag of M&M’s and select ONE, what is the probability that it is Blue or
Red? 15% + 20 % = 35% or probability of 0.35
i) Can the choice be Both Blue AND Red? NO!
ii) These sets are said to be:
P( B  R)  0
Mutually Exclusive? Not Mutually Exclusive
d) If we want to select one M&M, put it back and then select a 2nd M&M, what is the probability that they
will both be Blue? (0.15)(0.15) = 0.0225
Probability of two or more events occurring: P(A,B) = P(A) P(B|A)
e) By selecting WITH replacement (putting the first one back), we made these events:
INDEPENDENT
P(A,B) = P(A) P(B|A)… If independent, P(B|A) = P(B)
9
Classify each pair of events as not independent or independent.
16.
A shirt is chosen from a closet, then a scarf is chosen to match the shirt.
Since it has to match, then it is not independent.
17. A math book is randomly selected from a stack of math books, then a science book is randomly selected
from a stack of science books.
Independent
18. An odd-numbered problem is randomly chosen to count for a grade, then another odd-numbered problem
is selected.
NOT independent
19. Selecting a red M&M from a bag of M&M’s, then selecting a red skittle from a bag of skittles.
Independent
Are the following events independent or not independent?
Independent
20. Rolling a die and then flipping a coin
21. Flipping a coin and then flipping a coin
Independent
22. Picking a name out of a hat, not replacing it, and then picking another name out of the hat
NOT independent
23. Picking a card out of a deck, replacing it, then picking another card out of that deck
Independent
Are the following events mutually exclusive or not mutually exclusive?
24. Red cards and 7’s
Not Mutually Exclusive
7 of hearts and 7 of diamonds are both red.
25. Prime numbers and multiples of 2
Not Mutually Exclusive
2 is prime and a multiple of 2
26. Odd numbers and multiples of 6
Mutually Exclusive
Not Mutually Exclusive
You can take both the same semester
27. Taking Statistics and Pre-calculus
28. Make up a scenario that is independent. Make up a scenario that is mutually exclusive.
Example for independent: Pick a chocolate, then put it back and pick another
_______ chocolate. (with replacement)
Example of Mutually Exclusive: A card is picked from a deck of cards and a die is
___________________________ rolled.
10
Practice with Mutually Exclusive sets:
Integers from 1 to 20 are randomly selected. State whether the events are mutually exclusive.
29. Odd integers and multiples of 3
30. Integers less than 3 and integers greater than 15
NOT Mutually Exclusive
Mutually Exclusive
(3, 9, 15 are multiples of 3 AND ODD)
No overlap of two sets.
31. Odd integers and multiples of 2
32. Integers less than 20 and greater than 10.
Mutually Exclusive
NOT Mutually Exclusive
Probability of 2 events: P(A or B) = P(A) + P(B) – P(A and B)
Mutually Exclusive Events: If A and B are mutually exclusive, then they CANNOT happen at
the same time.
{Is there any overlap for mutually exlusive events? No. so P(A and B)
would = 0}
33. Find the probability of selecting a number from a set of 1-20 that is an odd integer and a multiple of 3?
(Make a Venn diagram…}
Mult 3
Odd
1 11
13
5
7
a)
P(Odd ) 
17
19
6
3
9
15
12
18
10
 0.5
20
b) P(Mult3)=
c) P(not Odd) =
P(notOdd )  1  P (odd )  1 
6
 0.3
20
c) P(Odd or Multiple of 3)=
10 6
3 13



20 20 20 20
10
 0.5
20
10
 0.5
20
3
d) P(Odd  Mult 3) 
13
P( Even) 
e) P(not Odd| Multiple of 3)
P (Odd | mult 3) 
11
3
 0.5
6
Calculating Probabilities:
Complements and Mutually Exclusive vs. Not Mutually Exclusive
1. Suppose that an event A has probability of
3
.
8
3 5
  0.625
8 8
What is P(A’)? _____________________________
1  P ( A)  1 
A’ means: Complement of A or NOT A.
2. Suppose that the probability of snow is 0.58. What is the probability that
it will NOT snow? 1-
0.58 = 0.42
A card is chosen from a well-shuffled deck of 52 cards. (4 suits, 13 of each suit…}
What is the probability that the card will be:
3. P(king
( K  Q)  P( K )  P(Q)  P( K  Q)
 queen)? P______________________
=
4. P(red jack
4
4
8
 0 
 0.1538
52 52
52
P( RJ  BQ)  P( RJ )  P( BQ)  P( RJ  BQ)
 black queen? _______________________
=
5. P(face
2
2
4
 0 
 0.0769
52 52
52
P(F  Prime)=P(F)+P(Prime)-P(F  Prime)
 card with a prime #)? ______________________
=
6. P(even
12 16 0 28
 

 0.5385
52 52 52 52
 Red)=P(Ev)+P(Red)-P(Ev  Red)
____________________
 red)? P(Even
=
7. P(spade
20 26 10 36
 

 0.6923
52 52 52 52
 J)=P(S)+P(J)-P(S  J)
 jack)? P(S
________________________
=
13 4 1 16
 

 0.3077
52 52 52 52
12
A spinner number 1-10 is spun. Each number is equally likely to be spun.
What is the probability of spinning:
P(E  Po3)=P(E)+P(Po3)-P(E  Po3)
5 3 0
8
=     0.8
10 10 10 10
8. P(even
 power of 3)?
____________________
9. P(odd
 power of 3)?
P(O  Po3)=P(O)+P(Po3)-P(O  Po3)
____________________
=
10. P(a number less than 8
5 3 3
5
    0.5
10 10 10 10
 a divisor of 15)?
_____________________
11. Look at the solution to the following problem and see if you can find the error (there definitely is a
mistake). Correct the error to find the right answer.
P(ace
 black) = P(ace) + P(black) =
4
26
30 15
+
=
=
52 52
52 26
P(A  B)=P(A)+P(B)-P(A  B)
Which of the problems (#1-11) above are about:
(write the problem number under its type)
COMPLEMENTARY events? MUTUALLY-EXCLUSIVE events? NOT MUTUALLY-EXCLUSIVE events?
1,2
3,4,5,8
6,7,9,10
GO back and check to see if you used the correct formula for each problem, based on its type. Make any
changes necessary.
13
Probability of Independent vs. Not Independent
Independent events
1. Bag A contains 9 red marbles and 3 green marbles. Bag B contains 9 black marbles and 6 orange marbles.
Find the probability of selecting one green marble from bag A and one black marble from bag B.
Independent: P(G  B)  P(G ) P( B)
=
3 9
 0.15
17 15
2. Two seniors, one from each government class are randomly selected to travel to Washington, D.C. Wes is in
a class of 18 students and Maureen is in a class of 20 students. Find the probability that both Wes and
Maureen will be selected.
Independent:
P(W  M )  P(W ) P( M )
1 1
=
 0.0028
18 20
3. If there was only one government class, and Wes and Maureen were in that class of 38 students, what
would be the probability that both Wes and Maureen would be selected as the two students to go to
Washington? Is this still an example of independent events?
Not Independent: P(W  M )  P(W ) P( M )
=
1 1
 0.0007
18 37
Not Independent Events
4. A box contains 5 purple marbles, 3, green marbles, and 2 orange marbles. Two consecutive draws are
made from the box without replacement of the first draw. Find the probability of each event.
(not independent)
a. P(orange  green)
2 3
P(O  G )  P (O ) P (G | O )
b. P(both marbles are purple)
10 9
 0.0667
P( P1  P2 )  P ( P1 ) P ( P1 | P2 )
c. P(purple  not purple)
P( P  P ')  P ( P ) P ( P ' | P )
5 4
 0.2222
10 9
5 5
 0.2778
10 9
5. If you draw two cards from a standard deck of 52 cards without replacement, find:
a. P(King
 Jack) P( K  J )  P( K )  P( J | K )  4 4  16  0.0060
52 51
b. P(face card
 ace) P( F  A)  P( F )  P( A | F )  12 4  4  0.0181
52 51
c. P(2 aces)
2652
P( A1  A2 )  P( A1 )  P( A2 | A1 ) 
14
221
4 3
1

 0.0045
52 51 221
MULTIPLE CHOICE:
6. A coin is tossed and a die with numbers 1-6 is rolled. What is P(heads
a. 1/12
b. 1/4
c. 1/3
 3)?
d. 2/3
7. Two cards are selected from a deck of cards numbered 1 – 10. Once a card is selected,
it is not replaced. What is P(two even numbers)?
a. 1/4
b. 2/9
c. 1/2
d. 1
8. Which of the following in NOT an example of independent events?
a.
b.
c.
d.
rolling a die and spinning a spinner
tossing a coin two times
picking two cards from a deck with replacement of first card
selecting two marbles one at a time without replacement
9. A club has 25 members, 20 boys and 5 girls. Two members are selected at random to
serve as president and vice president. What is the probability that both will be girls?
a. 1/5
b. 1/25
c. 1/30
d. 1/4
10. One marble is randomly drawn and then replaced from a jar containing two white marbles and one black
marble. A second marble is drawn. What is the probability of drawing a white and then a black?
a. 1/3
b. 2/9
c. 3/8
d. 1/6
11. Events A and B are independent. The P(A) = 3/5, and P(B’) = 2/3. What is P(A
a. 2/5
b. 1/5
c. 4/15
d. 2/15
P(B)=1-P(B’)
1
3 1
P ( A) P ( B ) 
5 3
15
 B)?
2 1

3 3
12.
Among the juniors and seniors in all of the AP Calculus BC classes, 65% are involved in Mathletes,
32% are involved in Scholastic Bowl, and 22% are involved in both.
Draw a Venn diagram to represent the situation. Hint: Start with the intersection/middle.
Math
0.43
Sch
0.22
0.10
If a person is randomly selected from the AP Calculus BC classes, find the probability of each:
a. The person is a Mathlete or is involved in Scholastic Bowl
0.43+0.22+0.10 = 0.75
b. The person is only involved in Mathletes.
OR
0.65 + 0.32 – 0.22 = 0.75
0.43
c. The person is only involved in Scholastic Bowl. 0.10
d. The person is neither a Mathlete nor is involved in Scholastic Bowl
P( M ' S ')  1  P ( M  S )
1  (0.43  0.22  0.10)  1  0.75  0.25
e. Are the events of being a Mathlete and involved in Scholastic Bowl independent? Why or why not?
No, the probability that a person is a mathlete would be different if it
was known that he/she was involved in scholastic bowl.
f. Are the events of being a Mathlete and involved in Scholastic Bowl disjoint? Why or why not?
No, a person can be a mathlete and be involved in scholastic bowl.
16
There are 8 blue marbles, 6 red marbles, 4 green marbles, and 2 yellow marbles in a bag.
13.
What is the probability of selecting a blue marble and then selecting a red marble? (by not stating that
the blue was replaced, we would assume that it is NOT replaced)
 8   6  48
 0.126
   
20
1
9
380
  
14.
What is the probability of selecting a blue marble, replacing it, and then selecting a red marble?
3
 8   6  48


 0.12
  
 20   20  400 25
Consider each probability.
15.
A coin is flipped and a number cube is rolled. What is the probability of getting tails and rolling a 4?
 1  1  1
 0.0833
   
2
6
12
  
Independent, not mutually exclusive
16.
There are 8 blue marbles, 6 red marbles, 4 green marbles, and 2 yellow marbles in a bag. What is the
probability of selecting two red marbles? total = 20
3
 6  5  30

 0.0789
   
20
19
380
38
  
NOT independent… The probability of Red changed.
Not mutually exclusive
17
One card is drawn from a 52-card deck. What is the probability of getting a black card and a heart?
There are no black cards that are hearts… 0
Mutually exclusive, so not independent.
18.
Go back to the previous 3 questions and determine if the events are independent and if they are mutually
exclusive.
17
Summarizing Probability Formulas
Given events A and B, find each formula.
P  A  B   P ( A) P ( B | A)
B
A
This is the probability of B, given that
A has already occurred. (IF A and B are independent, the
It won’t matter as P(B|A)=P(B)
P(A|B) =
I use algebra to manipulate the formula above….
P  A  B 
P( A | B) 
P( A  B)
P( B)
P( A)  P( B)  P( A  B)
B
A
P  A '  1 – P(A)
(called the complement of A)
P(Not A)
P  A '  B ' 
A
1  P( A  B)
A
P(At least one) = 1 – P(none)
18
B
Practice with the Formulas
19.
Two cards are drawn from a standard 52-card deck. Find the probability that a club and then a spade is
drawn.
P(C lub) P( Spade | C lub)
13 13 169

 .0637
52 51 2652
20.
One card is drawn from a standard 52-card deck. Find the probability that a king or a diamond is drawn.
4 kings, 13 diamonds, 1 King that is a diamond
P(king )  P( Diamond )  P( K  D)
4 13 1 16
 

 0.3077
52 52 52 52
21.
The probability that a student takes an AP class as a senior is 0.72, and the probability that a student
takes a science class as a senior is 0.56. The probability that a student takes an AP science class is 0.35.
Find the probability that a student takes an AP class or takes a science class.
P( AP  SCI )  0.35 P ( AP  Sci )  ?
P(AP)=0.72
P(sci)=0.56
P( AP  Sci )  P( AP)  P( Sci )  P( AP  Sci )
P( AP  Sci )  0.72  0.56  0.35  0.93
22.
If P  A  0.21, P  B   0.82 , and P  A  B   0.87 , find P  A  B  .
P( A  B )  P ( A)  P ( B )  P ( A  B )
0.87  0.21  0.82  P( A  B )
.16   P ( A  B )
P( A  B )  0.16
23. A natural number from 1 to 10 is randomly chosen. Find the probabilities:
a. P(even or 7)
b. P(even or odd)
P (even  Odd )
P ( Even)  P (Odd )  P( Even  Odd )
5 5
10
 0 
1
10 10
10
P(even  7)
P( Even)  P (7)  P( Even  7)
5 1
6 3
 0 
  0.6
10 10
10 5
c. P(multiple of 2 or multiple of 3)
P(2, 4, 6,8,10  3, 6,9)
5 3 1
7
    .7
10 10 10 10
d. P(odd or less than 3)
P (Odd   3)
P (Odd )  P ( 3)  P(Odd   3)
5 2 1
6 3
  
  0.6
10 10 10 10 5
19
Conditional Probabilities
Conditional probability is extremely important because it has a lot of real world applications; “If you stay out
too late, what is the probability that you will get grounded?, If you study for your test, what is the probability
that you will do well on your test? Or even, if you get sick with an illness, what is the probability that you will
need surgery?”
Yes
No
total
1. Sixty-three random people were asked their gender and
Male
16
12
28
if they wear their seatbelts regularly. The table shows
Female
20
15
35
the results of this survey. If a random person was
total
36
27
63
selected, find the probabilities below.
a. P (male)
b.
28 4
  0.4444
63 9
4
P(wears seatbelt regularly) 36
  0.5714
63 7
c. P(male
d. P(male
 does not wear seatbelt regularly) 11  0.1905
63
 does not wear seatbelt regularly) P( M )  P( SB)  P( M  SB)
e. P(male | does not wear seatbelt regularly)
28 27 12 43



 0.6825
63 63 63 63
12 4
  0.4444
27 9
f. Are gender and wearing a seatbelt independent? INDEPENDENT
4 16
P(M)?P(M|SB)
?
9 36
0.4444  .44444
2.
Real estate ads suggest that 64% of homes for sale have attached garages, 21% have swimming pools,
and 17% have both features. What is the probability that a randomly selected home for sale has:
a. A pool or a garage? 0.47+0.17+.04=
0.68
Pool
Garages
b. Neither a pool nor a garage? 1- 0.68 = 0.32
0.47
0.17
0.04
P( Pool ' G ')  1  P( Pool  G )
0.32
c. A pool but no garage?
P( Pool  G ')  0.04
d. Are having a pool and having a garage independent?
P(Pool)= 0.21
P(Pool|Garage)= 0.17 Not =
20
NO
II. Tree Diagrams
Tree diagrams can be helpful when finding the probability of multiple events using conditional probability.
3.
The diagram shows the probability that a day will begin clear or cloudy, and then the probability of rain
on days that will begin clear and cloudy. The path containing clear and rain represents days that you
believe will start out clear and then it actually rains.
a. Find the probability of it being a cloudy day.
0.72
b. Find the probability the day starting clear AND then raining.
P(Clear  Rain)  (0.28)(0.04)  0.0112
c. Find the probability of rain, given that the day began cloudy.
P( Rain | Cloudy ) 
P(rain  cloudy ) (0.72)(0.31)

 0.31
P(cloudy )
0.72
d. Find the probability that it will rain given it began clear.
P( Rain | Clear ) 
P(rain  clear ) (0.28)(0.04)

0.04
P(clear )
0.28
e. Find the probability that it will rain.
P( Rain)  P(Clear  Rain)  P(Cloudy  Rain)
= (0.28)(0.04) +(0.72)(0.31)
=0.2334
Properties of Tree Diagrams
1st branch: Simple
Probability
2nd branch: Conditional
Probability , so formula needed
Sum of probabilities of branches:
1
21
4. The tree diagram below shows the probability of school being cancelled when the temperature outside in
August is above and below 95°F.
0.7
a. Complete the missing probabilities on the diagram.
b. Find the probability the temperature is above 95°F and
there is no school.
0.72
Above
0.3
(0.72)(0.7)= 0.504
c. Find the probability that the temperature is below 95°F
and there is no school.
0.28
No School
School
0.1
No School
0.9
School
Below
(0.28)(0.1)= 0.028
d. Find the probability that there is no school.
P(above  no school) or P(below  no school)
0.504+ 0.028 = 0.532
e. Find the probability that there is school, given that the temperature is above 95°F.
P(school|above)=
P( school  above) (0.72)(0.3)

 0.3
P(above)
(0.72)
5. The ELISA test can help detect whether people have the HIV virus. As with any medical test, ELISA can
give false positive and false negative results. From experience, medical researchers estimate that the ELISA
test has a 0.2% false positive rate and a 0.1% false negative rate. Suppose the ELISA test is used in an area
where 5% of the population has HIV.
a. Draw a tree diagram to represent this situation.
0.999 Test +
0.05
0.95
HIV+
0.001
0.002
Test Test +
HIV-
0.998
Test -
b. Find the probability that a person does not have the virus and tests positive.
(False positive)  P( HIV  Test )
= (0.95)(0.002)=0.0019
c. Find the probability that a randomly selected person from this area would test positive.
(Positive)  P( HIV  Test )  P( HIV  Test )
= (0.05)(0.999) + (0.95)(0.002)=0.05185
d. Given that a randomly selected person has tested positive, find the probability that the person has HIV.
P(HIV+|Test+) 
P( HIV  Test ) (0.05)(0.999)

=0.9634
P(Test )
0.05185
22
5.
The students of a high school are 51% males; 45% of the males and 42% of the females attend concerts.
.45
M
.51
.55
.49
NC
C
.49
F
.51
NC
a. Find the probability that a student attends concerts.
47%
(.51)(.45) + (.49)(.49) = .4696
b. Find the probability that a student is a female and does not attend concerts.
(.49)(.51)= .2499
6.
Material
Paper
Metal
Glass
Plastic
Other
Total
Recycled
36.7
6.3
2.4
1.4
21.1
67.9
Not Recycled
45.1
11.9
10.1
24.0
70.1
161.2
You pick ONE item from a room, find the
81.8
 35.7%
229.2
67.9
 29.6%
P( picking out a recycled material)=
229.2
2.4
 1.05%
P( recycled and Glass) =
229.2
6.3
 9.28%
P(Metal | recycled)=
67.9
6.3
 34.6%
P( recycled | metal)=
18.2
1) P(picking out paper) =
2)
3)
4)
5)
23
Total
81.8
18.2
12.5
25.4
91.2
229.2
25%
7)
Male
In some
activity
120
Not in an
activity.
25
Female
150
40
270
65
a) Probability of being in some activity?
145
190
335
b) Probability of being a male?
270 54

 0.8060
335 67
145 29

 0.4328
335 67
c) Probability of being in some activity given you are a male?
120 24

 0.8276
145 29
8. A math teacher gave her class two tests. 80% of the class passed both tests and 95% of the class passed the
first test. What percent of those who passed the first test also passed the second test?
P( Pass  Pass )  P ( pass1st )( Pass 2nd | pass1st )
.80  0.95(2nd |1st )
P(2nd) =
0.8
 .842
0.95
Manipulating the Formula for Conditional Probability:
CONDITIONAL PROBABILITY FORMULA
For any two events A and B from a sample space with P(A) ≠ 0,
24
Assignment problems:
1) You are attending Pumpkinfest at Kipling, and decide to play a game. If you roll a 5 on a number cube, you
will win a pair of fake vampire teeth. You watch four people in front of you all roll a 5 and win! The
carnie tells you how lucky you are to be rolling the cube while it is on a hot streak. Your friend (who is
unfortunately not learning statistics with you), leans in and whispers in your ear, “You better not play.
You are pretty unlucky right now because there is no way this streak will continue”. Comment on both
of their statements.
This is using the LAW OF AVERAGES, which is not valid logic.
The probability will not change no matter what is rolled ahead of your roll.
2. Five multiple choice questions, each with four possible answers, appear on your history exam. Your nosey
partner knows that you just guessed on your exam and was surprised that you got all of the questions
correct. You partner said, “Good thing there wasn’t another question on that test, you would have been
due to get it wrong!” Comment on this person’s statement.
If you knew the material well enough to get them all correct, you knew
the material really well, so you would probably know the answer to the
next problem. This is using the law of averages that says that there is a
universal force that will equalize your score.
Integers from 1 to 100 are randomly selected. State whether the events in # 5-8 are mutually exclusive.
3. Even integers and multiples of 3
NOT Mutually Exclusive
6 , 12, 18… are in both sets.
4. Integers less than 40 and integers greater than 50
5. Odd integers and multiples of 4
Mutually Exclusive
Mutually Exclusive
6. Integers less than 50 and integers greater than 40 NOT Mutually Exclusive
41, 42,… are in both sets
7.
Explain what mutually exclusive means.
Two events that share no outcomes in common… also known as disjoint.
8.
Explain what independent means.
The out come of one trial does not influence or change the outcome of
another trial.
A test to check: P(A) = P(A|B)
25
9.
Can an event be both independent and mutually exclusive? Why or why not?
No, an event cannot be both independent and mutually exclusive because
if events are mutually exclusive, that means that when one event is
occurring, then the other on is not. If you know event A is true, does
that change the knowledge of event B? Yes, it cannot be true.
10.
Can an event be neither independent nor mutually exclusive? If not, why? If yes, can you give an
example?
Yes, it can be neither. Having long hair and being a male are neither
independent nor mutually exclusive. A person can be a male and have
long or short hair, so they are not mutually exclusive. Knowing that
someone has long hair will change the probability that that person being
a male, which makes the events not independent.
Classify each pair of events in # 9-12 as independent or not independent.
11. A member of the junior class and a second member of the same class are randomly selected.
Not Independent
12. A member of the junior class and a member of another class are randomly chosen.
Independent
13. Out of a box of t-shirts, you pick a medium and then a small.
Not Independent
14. You roll a number cube and get a five, then roll it again and get another five.
Independent
15.
Are “red card” and “spade” independent? Mutually exclusive?
Not Independent and Mutually Exclusive.
16.
Are “red card” and “ace” independent? Mutually exclusive?
Independent and Not Mutually Exclusive
17.
Are “face card” and “ace” independent? Mutually exclusive?
Not Independent and Mutually Exclusive.
26
Probability Rules:
Use proper notation and show all work to answer the following questions. Round probability to 3 decimals.
18.
In a recent health survey, randomly selected people were asked to check all of the following
statements that applied to them:
o
o
I run for exercise.
I swim for exercise.
The results were tabulated as follows: 35% said they were runners, 18% said they were swimmers, and
11% said they run and swim for exercise.
a. Draw a Venn diagram illustrating the results of this survey.
Swim
Run
0.11
0.24
0.07
0.58
b. What is the probability that a randomly selected person is a runner or a swimmer?
P( R  S )  P( R)  P( S )  P( R  S )
Prob of Runner OR Swimmer: 0.35+0.18-0.11=0.42
c. What is the probability that a randomly selected person is only a swimmer?
P( S  R ')  0.07
d. What is the probability that a randomly selected person is only a runner?
P( R  S ')  0.24
e. What is the probability that a randomly selected person is neither a runner nor a swimmer?
P( R ' S ')  1  P( R  S ) 
1- 0.42 = 0.58
19.
In recent survey of high school juniors asked what college entrance exam they plan on taking this year.
95% of the juniors surveyed stated that they will be taking the ACT or the SAT. 78% stated that they
will be taking at least the ACT. 63% stated that they will be taking at least the SAT.
a. What percentage of students are planning to take neither the ACT nor the SAT?
P( ACT ' SAT ')  1  P( ACT  SAT ) 
1- 0.95 = 0.05
b. What percentage of students will be taking both the ACT and the SAT?
P( ACT  SAT )  P ( ACT )  P ( SAT )  P ( ACT  SAT )
0.95=0.78+0.63- P( ACT  SAT )
0.95  0.78  0.63  P( ACT  SAT )
0.46= P( ACT  SAT )
27
20.
Suppose that 40% of cars in your area are manufactured in the United States, 30% in Japan, 10% in
Germany, and 20% in other countries. If cars are selected at random, find the probability that:
U.S
Japan
Germany
Other
0.4
0.3
0.1
0.2
a. A car is not U.S.-made.
b. It is made in Japan or Germany.
P(US ')  1  0.4  0.6
P( J  G)  P( J )  P(G)  P( J  G)
= 0.3+0.1-0=0.4
c. You see two in a row from Japan. (indep)
P ( J1  J 2 )  P ( J 1 ) P ( J 2 | J 1 )
d. You see a US car and then a German car.
Should say “You select a US car, then a German car”
P(US  G )  P (US ) P (G | US )
= (0.4)(0.1)=0.04
= (0.3)(0.3)=0.09
e. None of the three cars came from Germany.
P(G1 ' G2 ' G3 ')  P(G1 ') P(G2 ' | G1' ) P(G3 ' | G1'  G2 ')
= (0.9)(0.9)(0.9) = 0.729
21.
f. At least one of three cars is U.S.-made.
P(at least 1)= 1-P(none)
= 1- (0.4)(0.4)(0.4)= 0.784
Suppose that your soccer team has a box of jerseys for the game. You need to grab one for you and
your sister. You both need a medium shirt. The box contains 4 smalls, 10 mediums, and 6 larges. You
are quickly trying to grab your shirts and get to practice. (Assume that if you pick a wrong size, you do
not just put it back in the box, you set it aside). What is the probability that…
(a) You pick 2 smalls? P ( S  S )  P ( S ) P ( S | S )
1
2
1
2
1
 4  3  3
=    =  0.0316
 20  19  95
(b) You get the right shirts on the first two selects?
P( M 1  M 2 )  P(M 1 ) P(M 2 | M 1 )
 10  9  9
=    =  0.2368
 20  19  95
(c) You don’t get the first medium until the 4th shirt you choose?
P( M 1 ' M 2 ' M 3 ' M 4 )  P( M 1 ') P( M 2 ' | M 1 ') P( M 3 ' | M 1 ' M 2 ') P (M 4 | M 1 ' M 2 ' M 3 ')
 10  9  8  10  20
=      =
 0.0619
 20  19  18  17  323
22.
Five multiple choice questions, each with four possible answers, appear on your history exam. What is
the probability that if you just guess, you P(correct )  0.25
(a) get none of the questions correct? 
(b) get all of the questions correct? 
P(C ' C ' C ' C ' C ')  (0.75)5  0.2373
P(C  C  C  C  C )  (0.25)5  0.00098
(c) get at least one of the questions wrong?
1  P(none wrong)  1  (0.25)5  0.9990
28
23.
The probability that a student takes Spanish is 0.58, and the probability that a student takes French is
0.34. The probability that a student takes both languages is 0.06. Find the probability that a student
takes Spanish or French.
0.86
P  C   0.25 , P  D   0.64 , and P  C  D   0.71 , find P  C  D  .
0.18
24.
If
25.
One card is drawn from a standard 52-card deck. Find the probability that a five or a heart is drawn.
0.3077
26.
M and N are mutually exclusive events. Find P(M  N) if P ( M ) 
27.
A standard number cube is tossed. Find the probabilities:
28.
3
4
1
and P ( N )  .
6
a. P(even  3) 0.6667
b. P(less than 2  even)
c. P(prime  4) 0.6667
d. P(2  greater than 6) 0.1667
0.6667
Only 93% of the airplane parts that are examined pass inspection. What is the probability that at least
one of the next 3 parts pass inspection?
0.9997
1 – P(none) = 1- (0.07)3
29.
0.9167
Q and R are independent events. Find P(Q
 R) if P(Q) 
29
1
1
and P ( R )  . 0.05
4
5
30.
There is a 60% chance of thunderstorms the next three days.
a. What is the probability that there will be thunderstorms each of the next three days?
P(T  T  T )  P(T ) P(T ) P(T )
0.216
(0.6)3
b. What is the probability that it doesn’t rain at all over the next three days? 0.064
P(T ' T ' T ')  (0.4)3
c. What is the probability that it only rains on the third day? 0.096
P (T ' T ' T ) = P(N)
P(N) P(R)
d. What is the probability that it rains at least one of the three days? 0.936
P(at least 1) = 1 – P(none)
= 1 – P(no rain) = 1 – (0.4)3
31.
According to the American Pet Products Manufacturers Association (APPMA) 2003-2004 National Pet
Owners Survey, 39% of US households own at least one dog and 34% of US households own at least
one cat. Assume that 60% of US households own a cat or a dog.
a. What is the probability that a randomly selected person owns neither a cat nor a dog? 0.4
P( D ' C ')  1  P( D  C )
= 1 - 0.6 = 0.4
b. What is the probability that a randomly selected person owns both a cat and a dog? 0.13
P ( D  C )  P ( D )  P (C )  P ( D  C )
0.6 =0.39+0.34-P(D  C)
0.6-0.39-0.34=-P(D  C)
32.
A coin is flipped and a number cube is rolled. What is the probability of getting tails and rolling a 4?
0.0833
P(Tails  4)  P(T ) P(4 | T ) P(4|T)=P(4) since independent
1 1
=
2 6
33.
Two cards are drawn from a standard 52-card deck. Find the probability that a heart and then a
diamond is drawn.
0.0637
P( Heart  Diamond )  P( H ) P( D | H )
13 13
=
52 51
30
Conditional Probabilities
The table to the right gives information about the number of items of different materials that are recycled and
that are put in the garbage in a classroom. Mindy selects a random item from the bins. Find each probability.
34.
P(a glass item is selected) P (G ) 
0.1067
35.
36.
P(a recycled item is selected) P ( R )  80
150
0.5333
54
P(a paper item is selected) P ( P ) 
150
0.36
37.
38.
16
150
Material
Recycling Bin
Garbage
Bin
Paper
36
18
Glass
13
3
Plastic
23
7
Other
8
42
80
70
54
16
30
50
150
P(a paper item is selected from the recycling bin)
36
0.24
P( R  R) 
150
Is glass and being in the recycling bin independent? Not Independent
The table to the right shows the number of students that are in the orchestra and in chorus at a school.
One student is selected at random. Find each probability.
42.
P(student is male) 0.3889
70
P( M ) 
Male
180
P(student is in chorus) 0.6556
Female
118
P (C ) 
180
P(student is a male in the chorus) 0.2333
42
P( M  C ) 
 0.2333
180
P(student is male given the student is in chorus) 0.3559
43.
Are gender and musical class independent? Not Independent
39.
40.
41.
P(M | C )? P(M )
0.3559  0.3889
31
Orchestra
Chorus
28
42
34
76
62
118
70
110
180
44.
A survey is given to the graduating senior class. 67% of the students surveyed said that they took a
science class their senior year. 82% of the students said that they took a math class their senior year.
56% of the students said that they took both a math and a science class. Find the probability that a
student took a science class given that the student took a math class.
P( Sc | Ma) 
0.11
45.
0.56
0.26
P( Sc  Ma) 0.56

 0.6829
P( M )
0.82
A math teacher gave her class two tests. 80% of the class passed both tests and 95% of the class passed
the first test. What percent of those who passed the first test also passed the second test?
P(2nd |1st ) 
0.15 0.8
P(2nd 1st ) 0.8

 0.8421
P(1st )
0.95
46. The students at Deerfield High School chose their favorite activity.
The results are summarized in the table below.
Sports
Hiking
Reading
Computer Shopping
Other
Total
Female
39
48
85
62
71
29
334
Male
67
58
76
54
68
39
362
Total
106
106
161
116
139
68
696
One student is selected at random, find each probability.
a. P(sports)
106
 0.1523
696
c. P(sports|female)
e. P(reading
b. P(female)
39
 0.116766
334
d. P(female|sports)
 male) 76  0.109195
696
g. P(hiking|female)
334
 0.479885
696
f. P(male | reading)
48
 0.143713
334
h. P(hiking|male)
i. Are hiking and being a female independent? Not Independent
P(H|F) ? P(H)
48 106
?
334 696
0.1584  0.1437
32
39
 0.367925
106
76
 0.47205
161
58
 0.160221
362
47. The table contains information about the 1205 employees at one business. If one person is
selected at random, find the probabilities described below.
Education and Salary of Employees
Less than high
school
High school
Some college
College degree
total
Under
$20,000
$20,000 to
$30,000
Over
$30,000
total
69
36
2
107
112
102
13
296
98
193
178
505
14
143
245
404
224
438
436
1205
a. Find P(employee has less than a high school education).
b. Find P(employee earns under $20,000).
107
 0.088797
1205
296
 0.245643
1205
c. Find P(employee earns over $30,000 and has less than a high school education).
d. Find P(employee earns under $20,000 and has a college degree).
2
 0.00166
1205
13
 0.010788
1205
e. Given that the employee has a high school education, find the probability that the employee earns
14
 0.0625
over $30,000.
224
f. Given that the employee earns over $30,000, find the probability that the employee has at most a
16
 0.039604
high school education.
404
g. Are “some college” and “earning over $30,000” independent?
33
Not Independent
48.
About 35% of high school students participate on a varsity sport. Of those that participate on a varsity
sport, 8% go on to play a varsity sport in college. Of the HS students that do not play a varsity sport,
1% of those go on to play a sport in college.
a. Make a tree diagram for this situation.
.08
b. Find the probability that a randomly selected
college student plays a sport.
.35
HS
P (C )  P ( HS  C )  P ( HS ' C )
= P(HS)P(C|HS) +P(HS')(P(C|HS')
= (0.35)((0.08)+(0.65)(0.01)
.65
C
.92
C’
.01
C
.99
C’
HS’
= 0.0345
c. What is the probability that a randomly selected college student played a varsity sport in HS given
that you know this person plays a college sport?
P( HS | C ) 
P( HS  C ) (0.35)(0.08)

 0.8116
P(C )
(0.0345)
49. In a high school, 51% of the students are males. The students are polled, and 45% of the males and 49% of
the females say that they have attended at least 3 concerts in the past year.
a. Make a tree diagram for this situation.
.45
.51
C
b
Find the probability that a randomly selected student has
attended at least 3 concerts in the past year.47%
M
.55
.49
P(C)= (.51)(.45) + (.49)(.49) = .4696
NC
C
.49
F
.51
c.
Find the probability that a randomly selected student is a
female and attended less than 3 concerts in the past year.
.
25%
NC
(.49)(.51)= .2499
d.
Find the probability that a randomly selected student is
female, given that they have attended 3 or
more concerts that year.
34
0.5113
50.
32% of the senior class plays a competitive sport. 67% of the senior class is in a club. 26% of the senior
class are on a club and plays a competitive sport. If a randomly selected senior is chosen, what is the
probability that person is in a club, given that they play a competitive sport?
35
0.8125
Algebra II
Probability Review
1.
Name: ________________________
Multiple Choice: Which of the following are not independent events?
A.
B.
C.
D.
choice II only
choices II and III
choices I and III
choice III only
I. Getting an even number in the first and second roll of a number cube.
II. Getting an odd number when rolling a number cube and getting blue on a spinner.
III. Getting a face card in the first draw from a deck of playing cards, not replacing it, and then getting a face card in
the second draw.
2.
M and N are NOT mutually exclusive events. P(M) = 0.1 P(N) = 0.45 and P(M
Find P(M  N).
3.
Q and R are independent events. P(Q) = 0.8 and P(R) = 0.2. Find P(Q
 D) = 0.05.
 R).
In a class of 22 students, 10 study Spanish, 8 study French, and 5 study both French and Spanish.
4.
Make a Venn Diagram to represent the information:
5.
One student is picked at random. Find each probability.
a. P(studying Spanish but not French)
b. P(studying neither Spanish nor French)
c. P(studying both Spanish and French)
d. P(studying French)
e. What is the probability of a randomly selected student studying Spanish, given that they study French?
f. Are studying French and studying Spanish independent?
36
6.
The colors of M&M’s candies follow this distribution: 13% browns, 14% yellows, 13% reds, 24% blues, 20% oranges, and
16% greens.
a.
If you randomly select an M&M from a very large container, what is the probability that it is not green?
b.
If you randomly select an M&M from a very large container, what is the probability that it is blue or yellow?
c.
If you randomly select 3 M&M’s from a very large container without replacement, what is the probability that you get no
browns?
d.
If you randomly select 3 M&M’s from a very large container without replacement, what is the probability that you get at
least 1 orange?
7.
A small bag contains 4 red, 3 green, 2 yellow and 1 black M&Ms. If 2 are chosen at random find the probability that you
selected 2 red.
8.
Your messy sock drawer contains 6 brown socks, 5 black socks, and 9 white socks. If you blindly grab two socks (without
putting a sock back between grabs), what is the probability that you have at least one white sock?
9.
The probability that it is Friday and that a student is absent is 0.03. Since there are 5 days in a school week, the probability
that it is Friday is 0.2. What is the probability that a student is absent if it is Friday?
10.
A coin purse contains 4 pennies, 5 nickels, 8 dimes. Three coins are selected at random without replacements. Find the
probability that all three coins are dimes.
37
11.
From a standard deck of 52 cards, 4 cards are dealt. Find the probability that all four cards are red.
12.
From a standard deck of 52 cards, if one card is chosen at random, find the probability that it is a red card or a diamond.
13.
You are at a birthday party at Nickel City. Everyone at the birthday party gets to spin a wheel for extra tokens. The first
three people in line spin the wheel and land on “0 Tickets”. You are next in line. The kid behind you whispers to you, “Hey
– you sure are lucky to be next in line. You are due to get some bonus tickets!” Comment on this person’s statement.
14.
During spirit week, you made a note if people were dressed up and
what year in school they were. Here are your results in a table.
If you were to randomly select a person from your classes, what is
the probability that…
a.
The person is a Junior?
b.
The person is dressed up?
c.
The person is a senior wearing regular clothes?
d.
The person is a senior given that they are dressed up?
e.
Are spirit and class independent?
38
Junior
Dressed
Up
21
Regular
Clothes
15
Senior
Total
14
35
10
25
Total
36
24
60
15.
In a recent survey of athletes, 87% say that they do a warm-up before they begin working out. 65% say they do a cool-down
after they work out. 56% say they do both a warm-up and a cool-down.
a.
What is the probability that a randomly selected athlete does a warm up or a cool down with their workout?
b.
Are warm-ups and cool-downs independent?
Assume that about 10% of the people going through airport security have forgotten metal on them (didn’t take off their belt
buckle, change in their pocket, etc). Assume, also, that the scanners are 95% accurate (let’s hope that they are much more
accurate than that!!). Draw a tree diagram to help illustrate this situation.
16.
a.
If a randomly selected person goes through airport security, what is the probability that they will beep as positive (having
metal on them)?
b.
What is the probability that a randomly selected person has forgotten metal, given that they beeped as positive?
39