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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

Risk of a natural gas pipeline accident. Process Safety Progress (December 2004) published a risk analysis for a natural gas pipeline between Bolivia and Brazil. The most likely scenario for an accident would be natural gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (causing a jet fire) is .01. If the leak does not immediately ignite, it may result in a delayed ignition of a gas cloud. Given no immediate ignition, the probability of delayed ignition (causing a flash fire) is .01. If there is no ignition, the gas cloud will harmlessly disperse. Suppose a leak occurs in the natural gas pipeline. Find the probability that either a jet or flash fire will occur. Illustrate with a tree diagram.

Two fair dice are tossed, and the face on each die is observed.

  1. Use a tree diagram to find the 36 sample points contained in the sample space.
  2. Assign probabilities to the sample points in part a.
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A = {3showing on each die}

B = {Sum of two numbers showing is}

C = {Sum of two numbers showing is even}

An experiment results in one of three mutually exclusive events, A, B, or C. It is known that P (A)= .30, P(B)= .55 , and P(C)= .15. Find each of the following probabilities:

a. P(A鈭�B)

b.P(A鈭�C)

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d. P(B鈭�C)

e. Are B and C independent events? Explain.

Fuzzy logic in supply chain management. A branch of mathematics known as fuzzy logic was used to improve customer service in supply chain management. (Decision Analytics, February 2014.) Customers rate the importance of one service factor relative to another using the following numerical scale: 1 = service factors are equally important, 3 = one factor is moderately more important, 5 = one factor is strongly more important, 7 = one factor is very strongly more important and 9 = one factor is extremely more important. Fuzzy numbers were developed to allow for variation in customer responses. For example, the fuzzy number 1鈭紃epresents an actual response of either 1 or 3; the fuzzy number 7鈭紃epresents a response of 5, 7, or 9. Consider the probabilities of the actual responses for each fuzzy number shown in the table.

Fuzzy Response

Probabilities of Actual Responses

1~

P(1)=2/3,P(3)=1/3

3~

P(1)=1/3,P(3)=1/3,P(5)=1/3

5~

P(3)=1/3,P(5)=1/3,P(7)=1/3

7~

P(5)=1/3,P(7)=1/3,P(9)=1/3

9~

P(7)=1/3,P(9)=2/3

a. If a customer gives a fuzzy response7~, what is the probability that the actual response is not a 7?

b. If both5~9~ represent a possible fuzzy response of a customer, what are the possible actual responses for this customer?

The sample space for an experiment contains five sample points with probabilities as shown in the table. Find the probability of each of the following events:

a. Either 1,2 or 3 occurs

b. Either 1,3 or 5 occurs

c. 4 does not occur
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