There is a 1% delinquency rate for consumers with FICO (Fair Isaac and Company) credit rating scores above 800. If the Jefferson Valley Bank provides large loans to 12 people with FICO scores above 800, what is the probability that at least one of them becomes delinquent? Based on that probability, should the bank plan on dealing with a delinquency?


Here $X$ denote the number of consumers who becomes delinquents out of 12.

$p$ be the delinquency rate for customers with FICO credit rating scores above 800.

Given that $p=0.01$ and $n =12$. Thus $X\sim B(12, 0.01)$.

The probability mass function of $X$ is

$$ \begin{aligned} P(X=x) &= \binom{12}{x} (0.01)^x (1-0.01)^{12-x},\\ &\quad x=0,1,\cdots, 12 \end{aligned} $$

The probability that at least 1 of them becomes delinquent is

$$ \begin{aligned} P(X\geq 1) & =1- P(X=0)\\ &=1- \binom{12}{0} (0.01)^0 (1-0.01)^{12-0}\\ &= 1-0.8864\\ &= 0.1136 \end{aligned} $$

Based on that probability, the bank should plan on dealing with a delinquency because the probability is 0.1136 is greater than 0.05.

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