Question

The data in the table represents the breakdown of semiconductor wafers by lot and whether they conform to a thickness specification. If one wafer is selected at random,

1. what is the probability that the wafer conforms to specifications?
2. what is the probability that the wafer is from Lot A and conforms to specifications?
3. what is the probability that the wafer is from Lot A or conforms to specifications?
4. what is the probability that the wafer conforms to specifications, given the wafer is from Lot A?
5. Are the events “from Lot A” and “conforms to specifications” independent? Why or why not? Use the results from some of the previous parts of this exercise to answer part e.

Lots	Conforms to specifications	Does not conform to specifications	Totals
A	88	12
B	171	29
C	254	46
Totals

Answer 1

Table with totals included

Lots	Conforms to specifications	Does not conform to specifications	Totals
A	88	12	88+12=100
B	171	29	171+29=200
C	254	46	254+46=300
Totals	88+171+254=513	12+29+46=87	600

If a wafer is selected at random,

a. what is the probability that the wafer conforms to specifications?

Probability that the wafer conforms to specifications

=Total Number of wafers conforms to specifications/Total number of wafers = 513/600 = 0.855

Probability that the wafer conforms to specifications = 513/600 = 0.855

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b. what is the probability that the wafer is from Lot A and conforms to specifications

Probability that the wafer is from Lot A and conforms to specifications

=Total Number of wafers from Lot A and conforms to specifications (Cell value under : row: A and column : Conforms to specifications )/Total number of wafers = 88/600 = 0.1467

Probability that the wafer is from Lot A and conforms to specifications = 88/600 = 0.1467

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c. what is the probability that the wafer is from Lot A or conforms to specifications?

A : Event of a randomly selected wafer is from Lot A

B : Event of a randomly selected wafer conforms to specifications

probability that the wafer is from Lot A or conforms to specifications = P(A or B) = P(A)+P(B)-P(A and B)

From a. Probability that the wafer conforms to specifications = P(B) = 513/600

Probability that the wafer is from Lot A = Total number of wafers from Lot A / Total number of Wafers : P(A) = 100/600

From b. Probability that the wafer is from Lot A and conforms to specifications =P(A and B) = 88/600

P(A or B) = P(A)+P(B)-P(A and B) = 100/600 + 513/600 - 88/600 = 525/600=0.875

probability that the wafer is from Lot A or conforms to specifications = 525/600=0.875

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d. what is the probability that the wafer conforms to specifications, given the wafer is from Lot A?

Probability that the wafer conforms to specifications, given the wafer is from Lot A = P(B|A)

By Multiplication theorem,

P(A and B) = P(A)P(B|A)

P(B|A) = P(A and B)/P(A)

From b. Probability that the wafer is from Lot A and conforms to specifications =P(A and B) = 88/600

Probability that the wafer is from Lot A = Total number of wafers from Lot A / Total number of Wafers : P(A) = 100/600

P(B|A) = P(A and B)/P(A) = (88/600) / (100/600) = 88/100 = 0.88

Probability that the wafer conforms to specifications, given the wafer is from Lot A = 0.88

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e. Are the events "From lot A" and "Conforms to specifications" independent

A : Event of a randomly selected wafer is from Lot A

B : Event of a randomly selected wafer conforms to specifications

i.e are Event A and Event B independent;

As,

From a. Probability that the wafer conforms to specifications = P(B) = 513/600

From d. Probability that the wafer conforms to specifications, given the wafer is from Lot A = P(B|A) = 88/100

As P(B) P(B|A) ; A and B are not independent.

events "From Lot A" and "Conforms to specifications" are not independent