Question

In: Statistics and Probability

Bowl 1 contains 30 vanilla cookies and 10 chocolate cookies. What is the likelihood of choosing...

Bowl 1 contains 30 vanilla cookies and 10 chocolate cookies.

What is the likelihood of choosing 15 vanilla and 10 chocolate cookies (without replacement).

Now, instead, what is the likelihood of choosing 15 vanilla and 10 chocolate cookies (without replacement) if you had Bowl 2 which contains 20 of each.

Expert Solution

Lets first understand the concept of hypergeometric distribution to solve these two problems.

The hypergeometric distribution is used to calculate probabilities when selections are made from two groups without replacing members of the groups.

Hypergeometric Formula: Suppose a population consists of N items, k of which are successes. And a random sample drawn from that population consists of n items, x of which are successes. Then the hypergeometric probability is:

where

N: The number of items in the population.
k: The number of items in the population that are classified as successes.
n: The number of items in the sample.
x: The number of items in the sample that are classified as successes.

Bowl 1:

Bowl 1 contains 30 vanilla cookies and 10 chocolate cookies, then the likelihood of choosing 15 vanilla and 10 chocolate cookies without replacement is computed using hypergeometric formula.

In this context,

Total cookies in bowl, N = 30 (vanilla)+10 (chocolate) = 40
Let us consider getting vanilla cookies as getting success, then k = 30
We are drawing 15(vanilla) and 10(chocolate) cookies out of 40 cookies, so n=25
The number of vanilla cookies (success) in sample, x = 15

Then, the likelihood of choosing 15 vanilla and 10 chocolate cookies without replacement= P(X=15)

Answer: The likelihood of choosing 15 vanilla and 10 chocolate cookies without replacement is 0.003856

Bowl 2:

Bowl 1 contains 20 vanilla cookies and 20 chocolate cookies, then the likelihood of choosing 15 vanilla and 10 chocolate cookies without replacement is computed similarly we did above,

In this case,

Total cookies in bowl, N = 20 (vanilla)+20 (chocolate) = 40
Let us consider getting vanilla cookies as getting success, then k = 20
We are drawing 15(vanilla) and 10(chocolate) cookies out of 40 cookies, so n=25
The number of vanilla cookies (success) in sample, x = 15

Then, the likelihood of choosing 15 vanilla and 10 chocolate cookies without replacement= P(X=15)

Answer: The likelihood of choosing 15 vanilla and 10 chocolate cookies without replacement is 0.07121

orchestra answered 2 years ago

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