Answer:
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.
