Discrete Mathematics: Probability Exam
1.
Shade the given region on the corresponding Venn Diagram.
(a)
AC
U
B
A
C
(b)
AB
U
B
A
C
(c)
(A  B C)
U
B
A
C
(d)
B  C
U
B
A
C
(8pts)
2.
A school jazz band contains three different musical instruments – saxophone (S), clarinet (C)
and drums (D). Students in the band are able to play one, two or three different instruments.
In a class of 40 students, 25 belong to the jazz band. Out of these 25
4 can play all three instruments
5 can play the saxophone and clarinet
6 can play the clarinet and drums
8 can play the saxophone and drums
16 can play the saxophone
12 can play the drums.
(a)
Draw a Venn Diagram and clearly indicate the numbers in each region.
(5pts)
(b)
Find the number of students who play only the clarinet.
(2pts)
(c)
Find the probability that a student chosen at random from the class plays only the drums.
(2pts)
(d)
Find the probability that a student chosen at random from the class plays either the
clarinet or saxophone or both.
(2pts)
**
Given that a student plays the saxophone, find the probability that he also plays the
clarinet.
(Bonus!! 2pts)
2
3.
The following table shows the probabilities of certain outcomes on a loaded die.
Outcome (X)
probability
2
3
7
8
9
12
.12
.35
.06
.21
.15
.11
a) P(X < 5)
b) P(X > 8)
c) P(X = 10)
d) P(X< 7)
e) P(X = 8 | X >7)
f) P(X > 8)
(6pts)
4.
The table below shows the number of left and right handed tennis players in a sample of 50
males and females.
Left handed
Right handed
Total
Male
3
29
32
Female
2
16
18
Total
5
45
50
If a tennis player was selected at random from the group, find the probability that the player is
(a)
male and left handed;
(2pts)
(b)
right handed;
(2pts)
**** right handed, given that the player selected is female.
(2pts)
3
5.
100 students were asked which television channel (MTV, CNN or BBC) they had watched the
previous evening. The results are shown in the Venn diagram below.
U
MTV
CNN
35
19
23
6
5
2
3
7
BBC
From the information in the Venn diagram, write down the number of students who watched
6.
(a)
both CNN and MTV;
(2pts)
(b)
MTV or BBT;
(2pts)
(c)
CNN and MTV but not BBC;
(2pts)
(d)
BBC or CNN but not MTV.
(2pts)
Events A and B have probabilities P(A) = 0.4, P (B) = 0.65, and P(A  B) = 0.85.
(a)
Draw a Venn Diagram for this information.
(b) Calculate P(A  B).
(4pts)
(2pts)
4
7.
Marie has a bag of sweets which are all identical in shape. The bag contains 4 orange drops and
6 lemon drops. She selects on sweet at random, eats it, and then takes another at random.
Determine the probability that:
a) both sweets were orange drops
(2pts)
b) both sweets were lemon drops
(2pts)
c) the first was a lemon drop and the second was an orange drop.
(2pts)
d) the two sweets were different flavors
(2pts)
8. A restaurant has 2 appetizers, 4 salads, 8 entrees, and 4 desserts.
a. How many different meals can you create off the menu?
(2pts)
b. How many different meals can you make if you get your choice of an appetizer or salad, an
entrée and a dessert?
(2pts)
A bag of Lickety Split Marbles contains 6 red marbles, 6 green marbles, and 8 blue marbles. You are going to
draw out 2 marbles without replacement
9.
What is the probability that you will pull out a red marble then a green one?
(2pts)
10. What is the probability that you will pull out two red marbles?
(2pts)
11. What is the probability that you will pull out a green and a blue?
(3pts)
12. What is the probability that you will pull out a purple marble?
(2pts)
5
We are going to create passwords made of capital letters and numbers. The password must first contain 5
letters then 2 digits and then must end with either a letter or a number.
13. How many passwords are there?
(2pts)
14. How many passwords can be created if no character can be repeated?
(2pts)
15. How many passwords can be created if they cannot contain the digits 5 or 7.
(2pts)
16. How many passwords do not contain double characters? (no character can be beside itself, for example
EE or 55)
(2pts)
17. How many passwords contain at least one odd number?
(4pts)
18. 50 students went bushwalking, 23 were sunburnt, 22 were bitten by ants, and 5 were both sunburnt and
bitten by ants.
a. Draw a Venn Diagram for this situation
(4pts)
b. p(escaped being bitten)
(2pts)
c. p(either bitten or sunburnt)
(2pts)
d. p(neither bitten nor sunburnt)
(2pts)
e. p(had only one affliction)
(2pts)
6
In the senior class at Math I School students must take at least one of 3 math courses, either Statistics,
Calculus or Fractals. Here are the numbers for the senior class of 2007.
15 students take Statistics
35 students take Calculus
18 students take Fractals
9 students take both Statistics and Fractals
8 students take both Statistics and Calculus
5 students take both Calculus and Fractals
3 students take all three math courses.
19. Fill in the Venn Diagram with the appropriate counts
(8pts)
20. How many seniors attend Math I School?
(2pts)
21. Little Susie has given on going to college but her parents are forcing her to take the SAT. She decides
to bubble random answers for each question in a verbal section that has 24 questions. If each
question has 5 answer choices…
a. What is the probability of Susie getting exactly 15 out of the 24 questions correct?
b. What is the probability of Susie answering at least 18 correctly?
c. What is the probability of Susie answering at most 10 correctly?
d. What is the probability of Susie answering between 10 and 20 correctly?
7