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Chapter 3: Q.3.9 (page 110)

You ask your neighbor to water a sickly plant while you are on vacation. Without water, it will die with probability .8; with water, it will die with probability .15. You are 90percent certain that your neighbor will remember to water the plant.

(a) What is the probability that the plant will be alive when you return?

(b) If the plant is dead upon your return, what is the probability that your neighbor forgot to water it?

Short Answer

Expert verified

a). The probability that the plant will be alive when you return is 0.785.

b). The probability that your neighbor forgot to water it islocalid="1650015942117" 0.3721.

Step by step solution

01

 Given Information (Part a) 

W-. the plants were watered

PD鈭�Wc=0.8

P(D鈭�W)=0.15

P(W)=0.9

02

Explanation (Part a)

Formula of total probability can be applied here (because W鈭�Wc=鈭�,PW鈭�Wc=1 ):

P(D)=PD鈭�WcPWc+P(D鈭�W)P(W)

Formula for complement 鈬�PWc=1-P(W)=0.1

Substitution of familiar probabilities:

P(D)=0.8路0.1+0.15路0.9=0.215

Now formula for complement applied again:

PDc=1-P(D)=1-0.215=0.785

03

Given Information (Part b)

W-. the plants were watered

D- the plants died

Given probabilities:

PD鈭�Wc=0.8P(D鈭�W)=0.15P(W)=0.9
04

Explanation (Part b)

The definition of conditional probability gives:

PWc鈭�D=PWc鈭�DP(D)andPD鈭�Wc路PWc=PWc鈭�D

Which can be fused into:

PWc鈭�D=PD鈭�Wc路PWcP(D)

In part a) we computed PWc=0.1,P(D)=0.215is stated in the beginning, therefore:

localid="1650015923077" PWc鈭�D=0.8路0.10.215=1643鈮�0.3721

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Most popular questions from this chapter

Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve Cesarean (C) sections, and when a C section is performed, the baby survives 96 percent of the time when a C section is performed, the baby survives 96 percent of the time . If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives?

Consider a school community of mfamilies, with niof them having ichildren, i=1,鈥�,k,鈭�i=1kni=mConsider the following two methods for choosing a child:

1. Choose one of the mfamilies at random and then randomly choose a child from that family.

2. Choose one of the 鈭�i=1kinichildren at random.

Show that method 1is more likely than method 2to result

in the choice of a firstborn child.

Hint: In solving this problem, you will need to show that

鈭�i=1kini鈭�j=1knjj鈮�鈭�i=1kni鈭�j=1knj

To do so, multiply the sums and show that for all pairs i,j, the coefficient of the termninj is greater in the expression on the left than in the one on the right.

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, 铿乶d the conditional probability that

(a) the student is female given that the student is majoring in computer science;

(b) this student is majoring in computer science given that the student is female

Let A,B, and Cbe events relating to the experiment of rolling a pair of dice.

(a) If localid="1647938016434" P(A|C)>P(B|C)and localid="1647938126689" P(A|Cc)>P(B|Cc)either prove that localid="1647938033174" P(A)>P(B)or give a counterexample by defining events Band Cfor which that relationship is not true.

(b) If localid="1647938162035" P(A|C)>P(A|Cc)and P(B|C)>P(B|Cc)either prove that P(AB|C)>P(AB|Cc)or give a counterexample by defining events A,Band Cfor which that relationship is not true. Hint: Let Cbe the event that the sum of a pair of dice is 10; let Abe the event that the first die lands on 6; let Bbe the event that the second die lands on 6.

Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our disposal is a biased coin that lands on heads with some unknown probability p that need not be equal to 1 2 . Consider the following procedure for accomplishing our task: 1. Flip the coin. 2. Flip the coin again. 3. If both flips land on heads or both land on tails, return to step 1. 4. Let the result of the last flip be the result of the experiment.

(a) Show that the result is equally likely to be either heads or tails.

(b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?
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