A group of three undergraduate and five graduate students are available to fill

Question

A group of three undergraduate and five graduate students are available to fill certain student government posts. If four students are to be randomly selected from this group, (a) Find the probability that exactly two undergraduates will be among the four chosen. (b) What is the probability that at least two graduate students be chosen? [5] a. A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb lots. Let X = the number of lots ordered by a randomly chosen customer, and suppose that X has pmf Compute Expected Value of X, and what is the Standard Deviation of X? [5] b. Then compute the expected number of pounds left after the next customer's order is shipped, and the variance of the number of pounds left.

Explanation / Answer

10. A group of undergraduate...

a)

Let a success be obtaining an undergraduate.

There are 8 students here, 3 are undergraduate.

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    8      
K = number of successes in the population =    3      
n = sample size =    4      
x = number of successes in the sample =    2      
          
Thus,          
          
P(   2   ) =    0.428571429 [ANSWER]

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b)

Let a success be obtaining a graduate student.

There are 8 students here, 5 are graduate.

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative hypergeometric distribution table or technology, matching          
          
where          
N = population size =    8      
K = number of successes in the population =    5      
n = sample size =    4      
x = critical number of successes in the sample =    2      
          
Thus,          
P(at most   1   ) =    0.071428571
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.928571429 [ANSWER]

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