Question

Fraud detection has become an indispensable tool for banks and credit card companies to combat fraudulent credit card transaction
A fraud detection firm has detected some form of fraudulent activities in 1.31%, and serious fraudulent activities in 0.87%, of
transactions. Assume that fraudulent transactions remain stable.
a. What is the probability that in a given year, fewer than 2 out of 100 transactions are fraudulent? (Do not round intermediate
calculations. Round your final answer to 4 decimal places.)
o
Probability
b. What is the probability that in a given year, fewer than 2 out of 100 transactions are seriously fraudulent? (Do not round
intermediate calculations. Round your final answer to 4 decimal places.)
Probability

Answer 1

a. The probability that in a given year fewer than 2 out of 100 is 0.6226.

b. 0.7834 is the probability that in a given year, fewer than 2 out of 100 transactions are seriously fraudulent

How to solve for the biniomial distribution

Let the fraudulent transactions be = p = 0.0131

n = 100

This follows the binomial distribution

X~(n,p)

p(X = x) = 100Cx (0.0131)^x (1-0.0131)^100-x

When x < 2

100C0 * 0.0131⁰ (1-0.0131)¹⁰⁰ + 100 C1 * 0.0131¹ (1-0.0131)¹⁰⁰ ⁻¹

= 0.2675+0.3551

= 0.6226

The probability that in a given year fewer than 2 out of 100 is 0.6226.

b. x is the serious fraudulent transactions

0.87% = 0.0087n = 100

X~possion(λ = np)

λ = 100 * 0.0087

= 0.87

p(x<2) = p(x=0)+p(x=1)

$\frac{e^-^0^.^8^7(0.87)^0}{0!} + \frac{e^-^0^.^8^7(0.87)^1}{1!}$

= 0.4189 + 0.3645

= 0.7834

0.7834 is the probability that in a given year, fewer than 2 out of 100 transactions are seriously fraudulent.

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