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Question

# Suppose a computer chip manufacturer rejects 2% of the chips produced because they fail presale testing. What's the probability you find a bad one within the first 10 you examine?

Solution

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The distribution of a variable that measures the number of trials until the first success is a Geometric distribution.

Definition geometric probability:

$P(X=k)=q^{k-1}p$

We also know $p=2\%=0.02$ (reject) and $q=1-p=1-0.02=0.98$:

$P(X=1)=q^{k-1}p=0.98^{1-1}(0.02)=0.02$

$P(X=2)=q^{k-1}p=0.98^{2-1}(0.02)=0.0196$

....

$P(X=10)=q^{k-1}p=0.98^{10-1}(0.02)\approx 0.0167$

$P(X\leq 10)=P(X=1)+P(X=2)+...+P(X=10)=0.02+0.0196+...+0.0167=0.1829=18.29\%$

Command Ti83/84-calculator: geometcdf(0.02,10)

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