Question

Approximately 75% of people eat pasta at least once each week. What is the probability that for a randomly selected group of 16 people more than half of the people will eat pasta at least once in any randomly selected week? (Round to the nearest hundredth.)

Answer 1

Answer: the probability is 0.97

Explanation:

We want to determine the probability that more than 8 people will eat pasta at least once in any randomly selected week. We would apply binomial distribution formula,

P(x = r) = nCr × q^(n-r) × p^r

Where p = probability of success = 75/100 = 0.75

q is probability of failure = 1 - q = 1 - 0.75 = 0.25

n = number of sample = 16

P(x greater than 8) = 1 - P(x ≤ 8)

P(x ≤ 8) = P(x = 0) + P(x= 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7) + P(x = 8)

P(x = 0) = 16C0 × 0.25^(16-0) × 0.75^0 = 232.8 × 10^-12

P(x = 1) = 16C1 × 0.25^(16-1) × 0.75^1 = 0.00000001118

P(x = 2) = 16C2 × 0.25^(16-2) × 0.75^2 = 0.00000025146

P(x = 3) = 16C3 × 0.25^(16-3) × 0.75^3 = 0.0000035204

P(x = 4) = 16C4 × 0.25^(16-4) × 0.75^4 = 0.00003432389

P(x = 5) = 16C5 × 0.25^(16-5) × 0.75^5 = 0.00024713203

P(x = 6) = 16C6 × 0.25^(16-6) × 0.75^6 = 0.00136

P(x = 7) = 16C7 × 0.25^(16-7) × 0.75^7 = 0.0058

P(x = 8) = 16C8 × 0.25^(16-8) × 0.75^8 = 0.0197

P(x ≤ 8) = 0.027

P(x greater than 8) = 1 - 0.027 = 0.97