An urn contains 15 beads of which 7 are green and 8 are oran

An urn contains 15 beads of which 7 are green and 8 are orange. Let X be the    number of green beads selected in a sample of three beads. a. Calculate Pr ( X =2) if the sampling is done with replacement. b. Calculate Pr ( X =2) if the sampling is done without replacement. c. What is the name of the distribution that you used to answer (a)? d. What is the name of the distribution that you used to answer (b)? e. What is the name of the distribution you would use to calculate the probability that exactly five draws are needed to select 2 green beads if the sampling is done with replacement?  

Solution

A)

Here, the probability of getting green bead is constant, p = 7/30.

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    3      
p = the probability of a success =    0.233333333      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.125222222 [ANSWER]

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b)

If without replacement, we do a hypergeomtric distribution, with N = 30, K = 7 successes in the population, n = 3, k = 2 successes in the sample.

Thus,

P(x = 2) = [C(30-7, 1) C(7, 2)]/C(30, 3) = 0.118965517 [ANSWER]

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c)

Binomial distribution.

d)

Hypergeometric distribution.

e)

Negtaive binomial distribution.