Luke Skywalker is in the market for three new lightsabers. Yoda has a stash of t

Question

Luke Skywalker is in the market for three new lightsabers. Yoda has a stash of them, and in that stash, there are 5 blue lightsabers, 4 green lightsabers, 6 red lightsabers, and 3 white lightsabers. Luke only wants blue lightsabers, but Yoda informs him as follows: "A choice you have not. Choose at random you must". After each one chosen, Luke must keep it. How many possible ways can Luke choose 3 lightsabers and get at least two that are blue? What is the probability that Luke gets 3 blue lightsabers? Identify the distribution of the number of lightsabers Luke will get in his choosing of three. Be sure to identify all parameters of the distribution. What is the expected number of blue lightsebers Luke will get?

Explanation / Answer

a)

There are 18 lightsabers, 5 of them are blue.

Hence, there are 5C2= 10 ways to choose the first two to be blue, and 16 ways to choose the last one.

Hence, there are 10*16 = 160 ways. [ANSWER]

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b)

There are 18C3 = 816 ways to choose any 3 lightsabers.

There are 5C3 = 10 ways to choose all blue.

Hence,

P = 10/816 = 0.012254902 [ANSWER]

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c)

The distribution of blue lightsabers is a hypergeometric distribution with

P(x) = C(N-K, n-x) C(K, x) / C(N, n)          

N = population size = 18
K = number of successes (blue) in the population = 5
n = sample size = 3
x = the number of successes in the sample

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d)

E(x) = nK/N = 3*5/18 = 0.833333333 [ANSWER]

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