Your goal as this year\'s president of the Compost Club is to get 15 sororities

Question

Your goal as this year's president of the Compost Club is to get 15 sororities or fraternities to compost their food waste. Other schools' Composting Club presidents report that about 28% of sororities and fraternities will agree to participate and you expect that to be the case. You also expect the groups to respond independently of each other.

a) There are 50 sororities and fraternities on campus. You continue to ask sororities or fraternities to participate until you get 15 to commit. What is the likelihood that you will get 15 to participate before you've exhausted the list of sororities and fraternities?

b) Next, you've recruited help and now commit to asking all sororities and fraternities to participate, regardless of whether you reached your target of 15. Given that you've asked all sororities and fraternities to participate, what is the likelihood that 15 or more will participate?

Explanation / Answer

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then

probability mass function (pmf) of X is given by

p(x) = P(X = x) = (nCx)(px)(1 - p)n – x, x = 0, 1, 2, ……. , n ………………………..(1)

Now, to work out the solution,

Let X = Number of sororities or fraternities who agree to compost their food waste.

Then, X ~ B(50, 0.28) [Other schools' Composting Club presidents report that about 28% of sororities and fraternities will agree to participate and you expect that to be the case. => probability a sororite or fraternite will agree to compost their food waste = 0.28]

Part (a)

We want P(X = 15)

[using Excel Function], P(X = 15) = 0.1165 ANSWER

Part (b)

Here, we want P(X 15).

P(X 15) = 1 - P(X < 15) = 1 – 0.5714 = 0.4286 ANSWER [again using Excel Function]

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