49.Have you ever tried to get out of jury duty? About 25% of those called will f

Question

49.Have you ever tried to get out of jury duty? About 25% of those called will find an excuse (work, poor health, travel out of town, etc.) to avoid jury duty†. If 12 people are called for jury duty, find the following.

(a) What is the probability that all 12 will be available to serve on the jury? (Round your answer to three decimal places.)


(b) What is the probability that 6 or more will not be available to serve on the jury? (Round your answer to three decimal places.)


(c) Find the expected number of those available to serve on the jury. What is the standard deviation? (Round your answers to two decimal places.)

Explanation / Answer

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a.
P( X = 12 ) = ( 12 12 ) * ( 0.25^12) * ( 1 - 0.25 )^0
= 0.00000006

b.
P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 12 5 ) * 0.25^5 * ( 1- 0.25 ) ^7 + ( 12 4 ) * 0.25^4 * ( 1- 0.25 ) ^8 + ( 12 3 ) * 0.25^3 * ( 1- 0.25 ) ^9 + ( 12 2 ) * 0.25^2 * ( 1- 0.25 ) ^10 + ( 12 1 ) * 0.25^1 * ( 1- 0.25 ) ^11 + ( 12 0 ) * 0.25^0 * ( 1- 0.25 ) ^12
= 0.945597768
P( X > = 6 ) = 1 - P( X < 6) = 0.054402232

c.
Mean ( np ) =12 * 0.25 = 3
Standard Deviation ( npq )= 12*0.25*0.75 = 1.5

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