MAT 112, Finite Mathematics
Counting and Probability
Worksheet #17 solutions
1. Consider rolling a pair of dice. The possible outcomes are listed below.
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Based on this “sample space”, determine the following probabilities. Express the probability as a fraction. a). When the dice are rolled what is the probability that the sum of the numbers equals 9?
4 / 36 b). When the dice are rolled what is the probability that the sum of the numbers totals
7 or less?
21 / 36 c). When the dice are rolled what is the probability that one of the numbers rolled is a 5?
11 / 36 d). When the dice are rolled what is the probability that one of the numbers is a 3 or a 5?
20 / 36
e). When the dice are rolled what is the probability that either the sum equals 9 or one of
the numbers is a 5?
13 / 36
2. Suppose a 5-card hand is dealt from a deck of 52 playing cards. a). What is the probability of being dealt a hand of 5 spades?
C
13,5
= 0.0004952
C
52,5 b). What is the probability of being dealt a hand of 5 non-face cards spades?
C
40,5
= 0.25318
C
52,5 c). What is the probability of being dealt a hand including exactly 2 face cards?
C C
12,2 40,3
= 0.2509
C
52,5

3. Suppose a 13-card hand is dealt from a deck of 52 playing cards. a). What is the probability of being dealt a hand including exactly 5 spades?
C C
13,5 39,8
= 0.12469
C
52,13 b). What is the probability of being dealt a hand including exactly 4 or 5 spades?
C C
13,5 39,8

C C
13,4 39,9
= 0.12469 + 0.23861 = 0.3633
C
52,13 c). What is the probability of being dealt a hand including no spades?
C
39,13
C
52,13
= 0.01279
4. Thirty dogs have completed training to become “seeing-eye dogs”. The breed and gender
of the dogs is given in the following table.
German
Shepherd
Golden
Retriever
Labrador
Retriever
Total: female 4 male 6
8
4
5
3
Total: 10 12 8
Suppose one dog is selected at random. Determine the following probabilities
A). P (selected dog is female ) = 17 / 30
B). P (selected is female and German Shepherd ) = 4 / 30
C). P ( a female dog or a German Shepherd is selected ) = 23 / 30
D). P ( a German Shepherd or a Labrador Retriever is selected ) = 18 / 30
Suppose 10 of the dogs are selected at random to be transferred to Missouri.
Determine the following probabilities.
17
13
30
C C
17,5 13,5
E). P(the 10 selected dogs include 5 females and 5 males ) = = 0.26507
C
30,10
F). P(the 10 include 4 Labradors and 6 German Shepherds ) =
C C
8,4 10,6
C
30,10
= 0.000489
G). P(the selected dogs include exactly 4 Labradors or exactly 6 German Shepherds ) =
C C
8,4 22,6

C C
10,6 20,4

C C
8,4 10,6
= 6225660 / 30045015 = 0.2072
C
30,10