4. Chances: Risk and Odds in Everyday Life, by James Burke, reports that only 2%

Question

4. Chances: Risk and Odds in Everyday Life, by James Burke, reports that only 2% ofall local franchises are business failures. A Colorado Springs shopping complex has 130 franchises (restaurants, print shops, convenience stores, hair salons, etc.). What is the probability that: a) none of the franchises will be a business failure? (Write out formula) b) four or more franchises will be business failures? . An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. What is the probability that the first strike comes on the third well drilled? Find the mean and variance. 6. If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.

Explanation / Answer

QUESTION#4.
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
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 130 * 0.02
= 2.6
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 130 * 0.02 * 0.98
= 2.548
III.
standard deviation = sqrt( variance ) = sqrt(2.548)
=1.596246
ANSWER A.
P( X = 0 ) = ( 130 0 ) * ( 0.02^0) * ( 1 - 0.02 )^130
= 0.072342
ANSWER B.
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0) +   
= ( 130 3 ) * 0.02^3 * ( 1- 0.02 ) ^127 + ( 130 2 ) * 0.02^2 * ( 1- 0.02 ) ^128 + ( 130 1 ) * 0.02^1 * ( 1- 0.02 ) ^129 + ( 130 0 ) * 0.02^0 * ( 1- 0.02 ) ^130
= 0.736894
P( X > = 4 ) = 1 - P( X < 4) = 0.263106

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