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.
(B) There are 386 tickets, one for each student. Tickets are removed from 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: There are 386 tickets in total, 5 tickets are drawn without replacement, and the teacher is interested in predicting the number of winners from her class.
Understand the difference between probability distributions: The Binomial distribution is used when trials are independent and sampling is done with replacement, while the Hypergeometric distribution is used when sampling is done without replacement.
Recognize that in this problem, tickets are removed from the pool after being chosen, meaning the sampling is done without replacement. This is a key indicator that the Hypergeometric distribution is appropriate.
Recall the formula for the Hypergeometric distribution: \( P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}} \), where \( N \) is the population size, \( K \) is the number of successes in the population, \( n \) is the sample size, and \( k \) is the number of successes in the sample.
Conclude that the teacher should use the Hypergeometric distribution to model the number of winners from her class, as the problem involves sampling without replacement.
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