SOLUTION: Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that 12 people will be randomly chosen to be on the committee from a gr

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Question 1206790: Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that 12 people will be randomly chosen to be on the committee from a group of 31 volunteers, 21 who are technically proficient and 10 who are not. We are interested in the number on the committee who are not technically proficient.
How many instructors do you expect on the committee who are not technically proficient? (Round your answer to the nearest whole number.)
Find the probability that at least 3 on the committee are not technically proficient. (Round your answer to four decimal places.)
Answer by ikleyn(52855) About Me  (Show Source):
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Suppose that a technology task force is being formed to study technology awareness among instructors.
Assume that 12 people will be randomly chosen to be on the committee from a group of 31 volunteers,
21 who are technically proficient and 10 who are not. We are interested in the number on the committee who are not technically proficient.
(a) How many instructors do you expect on the committee who are not technically proficient?
(Round your answer to the nearest whole number.)
(b) Find the probability that at least 3 on the committee are not technically proficient.
(Round your answer to four decimal places.)
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This problem is on hypergeometric distribution.  N = 31 volunteers (total); n = 12 is the sample size, and 
k = 10 is the number of those who are not technically proficient (total).


(a)  The Math expectation for the number of those who are not technically proficient persons in all possible samples of 12 is

         E = %28n%2Ak%29%2FN = %2812%2A10%29%2F31 = 3.871,  or  4, when rounded to the nearest whole number.    ANSWER to (a).



(b)  To answer (b), it is easier to find the complementary probability first.

     The complementary probability is the probability to have LESS than 3 on the committee of 12 who are not technically proficient.


     P(less than 3 are not proficient) = P(0)+ P(1) + P(2) = %28C%5B10%5D%5E0%2AC%5B21%5D%5E12%29%2FC%5B31%5D%5E12 + %28C%5B10%5D%5E1%2AC%5B21%5D%5E11%29%2FC%5B31%5D%5E12 + %28C%5B10%5D%5E2%2AC%5B21%5D%5E10%29%2FC%5B31%5D%5E12 = 

                                        = %281%2A1%29%2F141120525 + %2810%2A21%29%2F141120525 + %2845%2A352716%29%2F141120525 = %281+%2B+210+%2B+15872220%29%2F141120525 = 0.1125  (rounded as requested).


     Then the probability that at least 3 are not proficient is                                                                                               

         1 - 0.1125 = 0.8875.    It is the ANSWER to (b)

Solved.

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As a reference, or as a source of information see this link

https://study.com/skill/learn/how-to-calculate-the-mean-or-expected-value-of-a-hypergeometric-distribution-explanation.html#:~:text=using%20the%20formula-,Using%20the%20values%20from%20step%201%20and%20the%20formula%20for,120%20%3D%20100%20120%20%3D%200.8333%20%E2%80%A6

or your textbook.