Binomial CDF vs PDF: Which to Use for "At Most 15 Students Play Sport" Probability?

Question

Exactly 25% of the students in a school play a sport. Fifty students are randomly selected to determine the probability that, at most, 15 students play a sport. Should a binomial probability density function or cumulative distribution function be used? Explain.

Accepted Answer

A binomial cumulative distribution function should be used because the question asks exactly 25% of students play a sport, to determine the probability that, at most, 15 students play a sport.

Answer

Step 1: Analyze the question's probability request: "at most 15 students play a sport" means we need the total probability that 0,1,2,...,15 students play a sport. Step 2: Recall the difference between binomial density and cumulative functions: The probability density function (PDF) calculates the probability of exactly k successes, while the cumulative distribution function (CDF) calculates the probability of k or fewer successes (the sum of PDFs from 0 to k). Step 3: Match the request to the function: Since we need the sum of probabilities for all values from 0 to 15, the cumulative distribution function is the correct choice.