A committee of three judges is randomly selected from among ten judges. Four of

Question

A committee of three judges is randomly selected from among ten judges. Four of the ten judges are tough; the committee is tough if at least two of the judges on the committee are tough. A committee decides whether to approve petitions it receives. A tough committee approves 50% of petitions and a committee that is not tough approves 80% of petitions (a) Find the probability a committee is tough (b) Find the probability a petition is approved (c) Suppose a petition can be submitted many times until it is approved. If a petition is approved with probability 3/4 each time, what is the mean number of times it has to be submitted until it is approved?

Explanation / Answer

(a) The total number of committees = 10C3 = 120

The number of tough committees = (4C2)(6C1)+(4C3) = 40

The probability of a tough committee is (4C26+4C3)/(10C3) = 40/120 = 1/3

(b) There is a 1/3 chance the comittee is tough, which correlates to a 50% chance of approval, and a 2/3 chance of a not tough comittee, which leads to a 80% chance of approval.

Hence, the probability of approval is  (1/3)(1/2)+(2/3)(4/5) = 1/6 + 8/15 = 0.70

(c) Probality of petition getting approved is 3/4 which means if a petition is submitted 4 times it will get approved 3 times. So, if in 1st attempt it gets rejected then in the 2nd attempt it will definitely get selected.

Hence, the mean number of times it has to be submitted until it is approved = 2.

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