MM1D2 a
MM1P1 a,b
MM1P2 b
MM1P3 a,b
MM1P4 c
The Addition Principle of Probability
Question:
Given two events A and B, what is the probability of the occurrence of either A or
B if the events are mutually exclusive? What is the probability of either A or B if
the events are not mutually exclusive? What are complementary events and how
are their probabilities related?
Launch:
Students are participating in a game of chance in their math class. Each student reaches
into a bag and pulls out one colored disk. Students cannot see the color of the disk until it
is pulled out. The bag contains a total of 25 disks. After each turn, the chosen disk is
placed back into the bag and the contents of the bag are mixed well before the next
student is allowed to choose. Below is a list of the colors of the disks, the numbers of
each color which can be found in the bag and the prize associated with each color of disk.
Color
Number Prize
White 15
Student receives nothing.
Red
7
Student receives 2 bonus points on next test.
Blue
2
Student receives 5 bonus points on next test.
Yellow 1
Student receives 10 bonus points on next test.
1)
What is the probability associated with each event described below?
Write each probability as a rational number and as a percentage.
Event W = student pulls out a white disk
P(W) = _________________
Event R = student pulls out a red disk
P(R) = _________________
Event B = student pulls out a blue disk
P(B) = _________________
Event Y = student pulls out a yellow disk
P(Y) = _________________
Event O = student pulls out an orange disk
P(O) = _________________
Event X = student pulls out a disk
P(X) = _________________
2)
Which event above is an impossible event and what is its probability?
3)
Which event above is a certain event and what is its probability?
Investigation:
Example 1:
Suppose there is a school club with the following membership roster. The roster contains
the first name, gender and class of each member.
First Name
Gender
Class
Beth
Female
Freshman
Caleb
Male
Freshman
John
Male
Freshman
Jorge
Male
Sophomore
Mary
Female
Sophomore
Lucy
Female
Sophomore
Abby
Female
Junior
Rob
Male
Junior
Sam
Male
Junior
Will
Male
Senior
Jada
Female
Senior
Joe
Male
Senior
Kelly
Female
Senior
A person in the club is to be chosen to travel to a national convention.
1.
How many students are there to choose from?
2.
How many seniors are there to choose from?
3.
How many juniors are there to choose from?
4.
Each student’s name is written on an individual piece of paper. Each piece is
folded and put into a jar. The selection of the student for the trip is made by a
random drawing of one name from the jar. Find the probability of each event
described below:
S = Event the chosen student is a senior
P(S) = _____
J = Event the chosen student is a junior
P(J) = ______
Definition:
Two events A and B are said to be disjoint or mutually exclusive if both A
and B cannot occur together in a single event. In such cases P(A and B) = 0.
5.
Are the events S and J described above mutually exclusive? Explain.
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6.
How many choices are there where the student is either a senior or a junior?
7.
Using your answer to question 6, determine the probability that the student
selected for the trip is either a senior or a junior.
8.
Describe the relationship between the answer to question 7 and the answers to
question 4.
The relationship discovered above is the Addition Principle of Probability for
Mutually Exclusive Events. It can be used to find the probability of the occurrence of
one event or another if the two events are mutually exclusive. Apply this principle to
answer the following question:
Question:
Suppose events A and B are mutually exclusive. The probability of event
A is 7/10 or .7 and the probability of event B is 2/10 or .2. What is the probability that
event A or event B occurs?
P(A or B) =
Now, fill in the blank in the following explanation of this principle.
Addition Principle of Probability for Mutually Exclusive Events:
Suppose events A and B are mutually exclusive. The probability of A is P(A). The
probability of B is P(B). Then the probability that either A or B will occur can be
denoted by P(A or B) and is equal to ______________________.
Example 2:
Definition: The complement of an event A (denoted by A ) is the event that A does
not occur. Events A and A are said to be complementary events.
1.
For the same problem as in Example 1 above, S was defined as the event that the
chosen student is a senior. Describe the complement S of event S?
2.
What is P( S )?
3.
Explain why S and S are mutually exclusive events?
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4.
Use the principle of probability for mutually exclusive events to determine P(S or
S ). This would be the probability that the chosen student either is or is not a
senior.
5.
Why does the value obtained in question 4 above seem to be a logical outcome for
this problem?
6.
Is it always the case that pairs of complementary events are mutually exclusive?
Explain.
Again, you have made a discovery! This time it is about the relationship between the
probabilities of complementary events! Use your discovery on the following question:
Question:
If event A has probability P(A) = .39, then what is the value of P( A )?
Fill in the blanks in the following:
For two complementary events A and A , the following are all true:
-
P(A) + P( A ) = ______
-
P(A) = 1 - ______
-
P( A ) = 1 - _______
Example 3: Again, refer to the problem from the previous examples. Suppose F is the
event that a female is chosen for the trip. Suppose S is the event that a senior is chosen
for the trip.
1.
Explain why events F and S would not be mutually exclusive.
2.
How many different ways can F occur? What is P(F)?
3.
How many different ways can S occur? What is P(S)?
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4.
How many different ways can both events F and S occur simultaneously? What is
the probability that the chosen student is both a female and a senior, P(F and S)?
5.
How many different ways can either event F or event S occur? What is the
probability that the chosen student is either a female or a senior, P(F or S)?
6.
What is the relationship between the probabilities P(F), P(S), P(F and S) and P(F
or S)?
This time, you have discovered a more general form of the addition principle of
probability from Example 1. This property applies to events even if they are not mutually
exclusive. Use what you have discovered to answer the following question:
Question:
There are two events A and B. You know that P(A) = .25, P(B) = .36 and
P(A and B) = .10. Determine P(A or B).
Fill in the blank in the following form of the addition principle of probability.
Addition Principle of Probability:
Suppose event A has probability P(A) and event B has probability P(B). The probability
that both A and B will occur is P(A and B). Then the probability that either A or B will
occur is P(A or B) = ______________________________.
The principle above is the most general version of the addition principle of probability
because it can be applied to problems where the events A and B are mutually exclusive as
well as to problems where they are not mutually exclusive. Explain why this form works
even when the events are mutually exclusive.
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Conclusions:
1.
In what type of situation is the addition principle of probability applied?
2.
What is the most general form of the addition principle of probability?
3.
What is the addition principle of probability for mutually exclusive events?
4.
What is the relationship between the probabilities of complementary events?
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In Class Problems:
1.
Suppose there is a spinner in the center of the circle shown below. The sectors of
the circle are all the same size, so the probability the spinner will land on each
value from 1 to 8 is equal.
1
8
7
2
6
3
5
4
a) For each event described below, list all numbers that would be possible spins
for that event. Then find the probability of the event:
Event E = the spinner lands on an even number
Event O = the spinner lands on an odd number
Event L4 = the spinner lands on a number less than 4
Event G5 = the spinner lands on a number greater than 5
Event M3 = the spinner lands on a multiple of 3
Event G8 = the spinner lands on a number greater than 8
b) Use the addition principle of probability to determine the following
probabilities:
P(E or O)
P(E or M3)
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P(L4 or G5)
P(O or G5)
c) Which pairs of events in part b) were mutually exclusive? Why?
d) Which pairs of events in part b) were complementary? Why?
e) Let G5 be the event that is the complement of event G5. Describe this event
and give its probability.
2.
Suppose some data was collected on some students from a high school. The
students were asked to choose one favorite fast food from among 3 choices and to
specify one favorite subject from among 4 choices. The table below is called a
contingency table. It shows the counts for each set of choices. For example, 45
students chose pizza as their favorite fast food and math as their favorite subject,
so 45 appears in the cell of the table in the Math row and the Pizza column.
Math
English
History
Science
Total
Pizza
45
32
36
28
Burgers
40
42
30
20
Mexican
30
34
30
16
Total
Grand Total =
a) Compute the row and column totals to fill in the blank cells in the table.
Make sure to add the row totals or the column totals (either will add to the
same value) to get the grand total. What does this value represent?
b) How many students chose pizza as their favorite fast food and math as their
favorite subject? What is the probability that a randomly chosen student
from the entire survey group would choose pizza and math as favorites?
c) How many students chose Mexican food as their favorite fast food? What is
the probability that a randomly chosen student from this group would
choose Mexican food as the favorite food?
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d) What is the probability that a randomly chosen student from this group
would choose pizza as the favorite food?
e) Are the events described in c) and d) mutually exclusive? Explain.
f) Use the addition principle of probability to find the probability that a
randomly chosen student from this group would choose either pizza or
Mexican food as the favorite fast food. Explain your result.
g) What is the probability a randomly chosen student would choose math as
the favorite subject?
h) Are the events of choosing math as the favorite subject and choosing
Mexican as the favorite fast food mutually exclusive? Why or why not?
i) Use the addition principle of probability to determine the probability that a
randomly chosen student from this group would choose either math as the
favorite subject or Mexican as the favorite fast food. Explain your result.
Closure:
Complete the following:
If two events A and B are mutually exclusive then _________________________
__________________________________________________________________
What is another term for mutually exclusive? ____________________________.
If event A has probability P(A) and event B has probability P(B) and it is known
that A and B are mutually exclusive then P(A or B) = _________________.
If event A has probability P(A) and event B has probability P(B) and the
probability that both A and B will occur is P(A and B) then
P(A or B) = _______________________.
If event A has probability P(A) then P( A ) = _______________.
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Homework:
1.
Barbara reaches into a bag of jelly beans and pulls one out without looking. The
bag contains 7 orange, 3 red, 6 green, 5 yellow and 2 black jelly beans. Find the
following probabilities:
a) Probability the jelly bean chosen is black
b) Probability the jelly bean chosen is red or green
c) Probability the jelly bean chosen is not yellow
2.
One month is chosen randomly from the twelve months of the year.
a) Find the probability of each of the following events:
Event J = the name of the month begins with the letter ‘J”
Event D = the name of the month begins with the letter “D”
Event BM = the month chosen comes before May
Event AA = the month chosen comes after April
b) Use the addition principle of probability to find the following probabilities:
P(J or AA)
P(J or D)
P(BM or AA)
P(D or AA)
c) Which pairs of events in part b) are mutually exclusive?
d) Which pair of events in part b) is complementary?
3.
Students were asked to choose their one favorite type or cake and one favorite
type of frosting from a short list of choices. The following contingency table shows the
results.
Chocolate
frosting
Cream cheese
frosting
Vanilla frosting
Total
Chocolate cake
35
Yellow cake
22
Carrot cake
0
17
32
12
20
23
8
Total
Grand Total =
a) Fill in the totals on the table.
b) Find the probability that a randomly chosen student who took this survey prefers
chocolate cake and vanilla frosting.
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c) Find the probability that a randomly chosen student who took this survey prefers
chocolate cake or vanilla frosting.
d) Find the probability that a randomly chosen student who took this survey prefers
cream cheese frosting or vanilla frosting.
e) Which frosting type and cake type were mutually exclusive events in this survey?
Explain.
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