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: the probability is 0.97

Step-by-step 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