Only 10% of American adults can name the capital of the Czech Republic. On a television game show the three randomly selected contestants are given the question "What is the capital of the Czech Republic?" The probability that none of them will know the answer is а bout: (a) 73% (b) 70% (c) 27% (d) 30% (e) 52% 5. A basketball player makes 70% of his free throws. If he is awarded 20 free throws, then the probability that he will make between 12 and 16 of them (including 12 and 16) is about (a).14 (b) 78 (c) 70 (d).53 (e).22

Problem 1: (a) 73%

Problem 2: (b) 78%

Explanation:

Problem 1:

The probability that a person can name the capital of Czech Republic is 10% which is 0.1. So,

p = 0.1

q = 1 - 0.1 = 0.9

n = 3

We will use the binomial distribution formula to find the probability that none of the contestants know the correct answer P (X=0). The binomial distribution formula is:

P (X=x) = ⁿCˣ pˣ qⁿ⁻ˣ

Where x = no. of successful trials

n = total no. of trials

p = probability of success

q = probability of failure

P (X=0) = ³C₀ (0.1) ⁰ (0.9) ³⁻⁰

P (X=0) = 0.729

0.729 x 100 = 72.9% ≅ 73% so option (a) is the correct answer.

Problem 2:

We will again use the binomial distribution formula to solve this problem. Here,

p = 70% = 0.7

q = 1 - 0.7 = 0.3

n = 20

We are required to find the probability P (12≤X≤16) which means we will compute P (X=12), P (X=13), P (X=14), P (X=15) and P (X=16) using the binomial distribution formula and then add them to find the final answer. So,

P (12≤X≤16) = P (X=12) + P (X=13) + P (X=14) + P (X=15) + P (X=16)

= ²⁰C₁₂ (0.7) ¹² (0.3) ²⁰⁻¹² + ²⁰C₁₃ (0.7) ¹³ (0.3) ²⁰⁻¹³ + ²⁰C₁₄ (0.7) ¹⁴ (0.3) ²⁰⁻¹⁴

+²⁰C₁₅ (0.7) ¹⁵ (0.3) ²⁰⁻¹⁵ + ²⁰C₁₆ (0.7) ¹⁶ (0.3) ²⁰⁻¹⁶

= 0.114 + 0.164 + 0.192 + 0.179 + 0.130

P (12≤X≤16) = 0.779

0.779 x 100 = 77.9% ≅ 78%

So, option (b) is the correct answer.