Probability Distributions - Extra Practice

Problem Context

A popular streaming platform reports that 0.35 of subscribers who watch a documentary’s introductory trailer add the full movie to their watchlist. A random sample of 40 subscribers who watched the trailer is selected. Let \(X\) represent the number of subscribers who add the documentary to their watchlist.

Practice Questions

Part 1

What is the exact probability distribution of \(X\)?

\(X \sim \text{Hypergeometric}(m = 40, n = 26, k = 14)\)
\(X \sim \text{Binomial}(n = 40, p = 0.35)\)
\(X \sim \text{Poisson}(\lambda = 14)\)
\(X \sim \text{Geometric}(p = 0.35)\)

Part 2

Which expression correctly computes the probability that exactly 12 subscribers add the documentary to their watchlist (\(P(X = 12)\))?

\(\binom{40}{12} (0.35)^{28} (0.65)^{12}\)
\((0.35)^{12} (0.65)^{28}\)
\(\binom{40}{12} (0.35)^{12} (0.65)^{28}\)
\(\binom{40}{12} (0.35)^{12} (0.35)^{28}\)

Show Solution

Solution & Explanations

Part 1

The situation fits a Binomial Distribution because it satisfies the four core requirements:

Binary: Each subscriber either adds the movie (“success”) or does not (“failure”).
Independent: One subscriber’s choice does not change another subscriber’s probability.
Number of trials is fixed (\(n = 40\)).
Success probability remains constant (\(p = 0.35\)).

Thus, the correct shorthand notation is: \(X \sim \text{Binomial}(n = 40, p = 0.35)\)

Part 2

To find the probability of exactly \(k\) successes, use the binomial probability mass function (PMF) formula:

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Substituting our known problem parameters:

Total trials (\(n\)) = \(40\)
Target successes (\(k\)) = \(12\)
Probability of success (\(p\)) = \(0.35\)
Probability of failure (\(1 - p\)) = \(1 - 0.35 = 0.65\)

Placing these directly into the formula gives us the unsimplified expression:

\[\binom{40}{12} (0.35)^{12} (0.65)^{28}\]