5. Binomial Distribution & Discrete Random Variables
Hypergeometric Distribution
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A school is holding a fair raffle and a teacher is interested in predicting how many winners will be from her class. Determine which probability distribution she should use given the following information.
(A) There are 386 tickets, one for each student. Tickets are placed back in the pool after being chosen and 5 tickets are drawn.
A
Binomial
B
Hypergeometric
Verified step by step guidance1
Identify the key characteristics of the problem: The teacher is interested in predicting the number of winners from her class. The problem specifies that tickets are placed back in the pool after being chosen, which means the draws are independent and the probability of success remains constant for each draw.
Recall the conditions for a Binomial distribution: (1) There are a fixed number of trials (n), (2) Each trial has only two possible outcomes (success or failure), (3) The probability of success (p) is constant for each trial, and (4) The trials are independent. Verify that these conditions are met in the problem.
Understand why the Hypergeometric distribution is not appropriate: The Hypergeometric distribution is used when sampling is done without replacement, meaning the probability of success changes after each draw. Since the problem specifies that tickets are placed back in the pool after being chosen, this condition does not apply.
Conclude that the Binomial distribution is the correct choice: Since the draws are independent, the probability of success is constant, and there are a fixed number of trials (5 tickets drawn), the Binomial distribution is the appropriate model for this scenario.
To apply the Binomial distribution, define the parameters: Let n = 5 (number of trials), p = probability of selecting a ticket from the teacher's class (this would depend on the number of students in her class divided by the total number of tickets, 386), and X = number of winners from her class. The probability mass function for the Binomial distribution is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where k is the number of successes.
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