## Related questions with answers

A population of 600 semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification. The following table presents the number of wafers in each category. A wafer is chosen at random from the population.

$\begin{matrix} \text{Lot} & \text{Conforming} & \text{Nonconforming}\\ \text{A} & \text{88} & \text{12}\\ \text{B} & \text{165} & \text{35}\\ \text{C} & \text{260} & \text{40}\\ \end{matrix}$

a. If the wafer is from Lot A, what is the probability that it is conforming? b. If the wafer is conforming, what is the probability that it is from Lot A? c. If the wafer is conforming, what is the probability that it is not from Lot C? d. If the wafer is not from Lot C, what is the probability that it is conforming?

Solution

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Let $D=$Conforming

(a) 88 of the $88+12=100$ wafers from Lot A are conforming.

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P(D|A)=\dfrac{\text{\# of favorable outcomes}}{\text{\# of possible outcomes}}=\dfrac{88}{88+12}=\dfrac{88}{100}=\dfrac{22}{25}\approx 0.88$

(b) 88 of the $88+165+260=513$ conforming wafers are from Lot A.

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P(A|D)=\dfrac{\text{\# of favorable outcomes}}{\text{\# of possible outcomes}}=\dfrac{88}{88+165+260}=\dfrac{88}{513}\approx 0.1715$

(c) $88+165=253$ of the $88+165+260=513$ conforming wafers are not from Lot C.

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P(C^c|D)=\dfrac{\text{\# of favorable outcomes}}{\text{\# of possible outcomes}}=\dfrac{88+165}{88+165+260}=\dfrac{253}{513}\approx 0.4932$

(d) $88+165=253$ of the $88+12+165+35=300$ wafers not from Lot C are conforming.

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P(D|A)=\dfrac{\text{\# of favorable outcomes}}{\text{\# of possible outcomes}}=\dfrac{88+165}{88+12+165+35}=\dfrac{253}{300}\approx 0.8433$

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