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Chapter 3: Q3-55E (page 193)

For two independent events, A and B, P (A) = .4 and P(B) = .2 :

a. Find P (A鈭〣)

b. Find P (A/B)

c. Find P (A鈭�B)

Short Answer

Expert verified

Answer

  1. 0.08
  2. 0.4
  3. 0.52

Step by step solution

01

Step-by-Step SolutionStep 1: Introduction

There are two occurrences, A and B. If the outcome of the A event has no impact on the outcome of the B, the two occurrences are said to be independent.

P(A鈭�B)= P(A)脳P(B)

02

Find the required probability

Since A and B are independent events.

Therefore,

P (A鈭�B) = P(A)脳P(B)= .4脳.2= .08

Hence, the required probability is 0.08.

03

Find the required probability

P (A/B) =P (A鈭�B)P (B)=.08.2=.4

Hence, the required probability is 0.4.

04

Find the required probability

P (A鈭�B) = P(A) + P (B)鈭�P(A鈭�B)= 0.4 + 0.2鈭�0.08= 0.52

Hence, the required probability is 0.52.

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Most popular questions from this chapter

Jamming attacks on wireless networks. Refer to the International Journal of Production Economics (Vol. 172, 2016) study of U.S. military jamming attacks on wireless networks used by terrorists, Exercise 2.8 (p. 73). Recall that 80 recent jamming attacks were classified according to network type (WLAN, WSN, or AHN) attacked and the network's number of channels (single- or multi-channel). The results are reproduced in the accompanying table.

a. Find the probability that a recent jamming attack involved a single-channel network.

b. Find the probability that a recent jamming attack involved a WLAN network.


Network Type/Number of Channels

Number of Jamming Attacks

WLAN / Single

31

WSN / Single

13

AHN / Single

8

WLAN / Multi

14

WSN / Multi

9

AHN / Multi

5

TOTAL

80

Source: S. Vadlamani et al., "Jamming Attacks on Wireless Networks: A Taxonomic Survey, "International Journal of Production Economics, Vol. 172, 2016 (Figure 6)

Consider the experiment depicted by the Venn diagram, with the sample space S containing five sample points. The sample points are assigned the following probabilities:

P (E1) = .20, P (E2) = .30, P (E3)= .30, P (E4) = .10, P (E5) = .10.

a. Calculate P (A), P (B), and P (A鈭�B).

b. Suppose we know that event A has occurred, so that the reduced sample space consists of the three sample points in A鈥攏amely, E1, E2, and E3. Use the formula for conditional probability to adjust the probabilities of these three sample points for the knowledge that A has occurred [i.e., P (Ei/A)]. Verify that the conditional probabilities are in the same proportion to one another as the original sample point probabilities.

c. Calculate the conditional probabilityP (E1/A)in two ways: (1) Add the adjusted (conditional) probabilities of the sample points in the intersection A鈭�B, as these represent the event that B occurs given that A has occurred; (2) use the formula for conditional probability:

P (B/A) =P (A鈭�B)P (A)

Verify that the two methods yield the same result.

d. Are events A and B independent? Why or why not?

Ranking razor blades.The corporations in the highly competitive razor blade industry do a tremendous amount of advertising each year. Corporation G gave a supply of three top-name brands, G, S, and W, to a consumer and asked her to use them and rank them in order of preference.

The corporation was, of course, hoping the consumer would prefer its brand and rank it first, thereby giving them some material for a consumer interview advertising campaign. If the consumer did not prefer one blade over any other but was still required to rank the blades, what is the probability that

a.The consumer ranked brand G first?

b.The consumer ranked brand G last?

c.The consumer ranked brand G last and brand W second?

d.The consumer ranked brand W first, brand G second, and brand S third?

Simulate the experiment described in Exercise 3.7 using any five identically shaped objects, two of which are one colour and the three another colour. Mix the objects, draw two, record the results, and then replace the objects. Repeat the experiment a large number of times (at least 100). Calculate the proportion of time events A, B, and C occur. How do these proportions compare with the probabilities you calculated in Exercise 3.7? Should these proportions equal the probabilities? Explain.

Chance of an Avon sale.The probability that an Avon salesperson sells beauty products to a prospective customer on the first visit to the customer is .4. If the salesperson fails to make the sale on the first visit, the probability that the sale will be made on the second visit is .65. The salesperson never visits a prospective customer more than twice. What is the probability that the salesperson will make a sale to a particular customer?
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