Section 6.1-An Introduction to Discrete Probability-Discrete Mathematics and Its Applications - Part 3

Rosen, Discrete Mathematics and Its Applications, 6th edition

Extra Examples
Section 6.1—An Introduction to Discrete Probability (Part - 3)

p.397, icon at Example 9

#1. Suppose S = {1, 2, . . . , 20}. You select a subset T ⊆ S of size three. Find the probability that T has at least one even number in it.

Solution:

There are C(20, 3) subsets of size three, and choosing any of them is equally likely. It is easiest to use the rule p(E) = 1 − p(E). Let E be the event “T has at least one even number in it”. Therefore E is the event “T has only odd numbers in it”. We have 

p(E) = 1 − p(E) = 1 − C(10, 3) / C(20, 3) ≈ 0.895.

p.397, icon at Example 9

#2. Suppose S = {1, 2, . . . , 20}. You select a subset T ⊆ S of size three. Find the probability that T contains the numbers 10 or 20.

Solution:

There are C(20, 3) subsets of size three, and choosing any of them is equally likely. We use the rule for finding the probability of the union of two events, where E is the event “10 ∈ T” and F is the event “20 ∈ T ”. Note that we must subtract p(E ∩ F) because both 10 and 20 might be elements of T .

p(E ∪ F) = p(E) + p(F) − p(E ∩ F)

 = C(19, 2)/C(20, 3) + C(19, 2)/C(20, 3) − C(18, 1)/C(20, 3)

= C(19, 2) + C(19, 2) − C(18, 1)/C(20, 3)

≈ 0.284.

p.397, icon at Example 9

#3. A true/false quiz has 10 questions. If you randomly answer each question, what is the probability that

you score at least 70%?

Solution:

To score at least 70%, you need to answer 7, 8, 9, or 10 questions correctly. There is C(10, 10) = 1

way to answer all ten questions correctly, C(10, 9) = 10 ways to correctly answer nine questions correctly,

C(10, 8) = 45 ways to answer eight questions correctly, and C(10, 7) = 120 ways to answer seven questions correctly. Thus, the probability of answering at least seven questions correctly is

p(answer at least 7 correctly) = p(answer 10 correctly) + p(answer 9 correctly) + p(answer 8 correctly) + p(answer 7 correctly) 

=  1/ 2^10 + 10/2^10 + 45/2^10 + 120/2^10

= (1 + 10 + 45 + 120)/210 

= 176/1024

≈ 0.172.