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Chapter 3: Q81E (page 200)

Suppose the events B1and B2are mutually exclusive and complementary events, such thatP(B1)=.75andP(B2)=.25 Consider another event A such that role="math" localid="1658212959871" P(AB1)=.3, role="math" localid="1658213029408" P(AB2)=.5.

  1. FindP(B1鈭�A).
  2. FindP(B2鈭�A)
  3. Find P(A) using part a and b.
  4. Findrole="math" localid="1658213127512" P(B1A).
  5. Findrole="math" localid="1658213164846" P(B2A).

Short Answer

Expert verified

The values of probabilities in all parts are:

  1. 0.225
  2. 0.125
  3. 0.35
  4. 0.64
  5. 0.357

Step by step solution

01

Important formula

The general rule of multiplication is P(A鈭�B)=P(A)脳P(BA).

02

Find P(B1∩A).

P(B1鈭�A)=P(AB1).P(B1)=(0.3)(0.75)=0.225

So, the values of probabilities in all parts are 0.225

03

Find P(B2∩A)

P(B2鈭�A)=P(AB2).P(B2)=(0.25)(0.5)=0.125

Hence, the values of probabilities in all parts are 0.125

04

Find P(A)

P(A)=P(B1鈭�A)+P(B2鈭�A)=0.225+0.125=0.35

Accordingly, the values of probabilities in all parts are 0.35

05

Find P(B1A)

P(B1A)=P(B1鈭�A)P(A)=0.2250.35=0.64

Henceforth, the values of probabilities in all parts are 0.64

06

Find P(B2A)

P(B2A)=P(B2鈭�A)P(A)=0.1250.35=0.357

Therefore, the values of probabilities in all parts are 0.357

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Most popular questions from this chapter

Confidence of feedback information for improving quality. In the semiconductor manufacturing industry, a key to improved quality is having confidence in the feedback generated by production equipment. A study of the confidence level of feedback information was published in Engineering Applications of Artificial Intelligence(Vol. 26, 2013). At any point in time during the production process, a report can be generated. The report is classified as either 鈥淥K鈥� or 鈥渘ot OK.鈥� Let Arepresent the event that an 鈥淥K鈥� report is generated in any time period (t).Let Brepresent the event that an 鈥淥K鈥� report is generated in the next time period. Consider the following probabilities:

P(A)=0.8,PBA=0.9, andPBAC=0.5.

a. Express the event B|Ain the words of the problem.

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g. Use Bayes鈥� Rule to find P(A|B), i.e., the probability that an 鈥淥K鈥� report was generated in one time period(t), given that an 鈥淥K鈥� report is generated in the next time period(t+1).

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Are events A and B independent? Why?

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