5.88 Bad checks. Duck Worth Wearing, a children’s consignment shop, accepts cash

Question

5.88 Bad checks. Duck Worth Wearing, a children’s consignment shop, accepts cash,check, and credit card for payment. About 30% of all transactions are paid by check.Duck Worth Wearing anticipates only 3 bad checks per month, on average. Assume that the number of bad checks per month can be modeled as a Poisson random variable.

(a) What is the probability of no bad checks in the next month?

(b) What is the probability of one or more bad checks in the next month?

(c) What is the probability of more than 7 bad checks in the next month?

Explanation / Answer

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a)
P( X = 0 ) = e ^-3 * 3^0 / 0! = 0.0498
b)
P( X < 1) = P(X=0)
= e^-3 * 0 ^ 0 / 0!   
= 0.0498
P( X > = 1 ) = 1 - P (X < 1) = 0.9502
c)
P( X < = 7) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-3 * 3 ^ 7 / 7! + e^-3 * 0 ^ 6 / 6! + e^-3 * ^ 5 / 5! + e^-3 * ^ 4 / 4! + e^-3 * ^ 3 / 3! + e^-3 * ^ 2 / 2! + e^-3 * ^ 1 / 1! + e^-3 * ^ 0 / 0!
= 0.9881
P( X > 7) = 1 -P ( X <= 7) = 1 - 0.9881 = 0.0119

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