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Chapter 3: Q. 3.23 (page 99)

Urn I contains 2white and 4red balls, whereas urn II contains 1white and 1red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II. What is

(a) the probability that the ball selected from urn II is white?

(b) the conditional probability that the transferred ball was white given that a white ball is selected from urn II?

Short Answer

Expert verified

a)The probability that the ball selected from urn II is white is 49

b) white ball is selected from urn II islocalid="1649413182090" 12

Step by step solution

01

Step1:introduction

Let Wi,i鈭�[1,2]be the event that a white ball is drawn from urnlocalid="1649493600869" Iand Ribe the event that a red ball is drawn from urn i,i鈭�[1,2]

02

Step2:The ball selected from urn II is white 

PW2=PW2鈭�W1PW1+PW2鈭�R1PR1=23脳26+13脳46=49

03

A white ball is selected from urn II 

PW1鈭�W2=PW2鈭�W1PW1PW2=23脳2649=12

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Most popular questions from this chapter

Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer the question, pointing out that if A knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from 1 3 to 1 2 because he would then be one of two prisoners. What do you think of the jailer鈥檚 reasoning?

The king comes from a family of 2 children. What is the probability that the other child is his sister?

There is a 60percent chance that event Awill occur. If Adoes not occur, then there is a 10percent chance that Bwill occur. What is the probability that at least one of the events AorBwill occur?

Let S = {1, 2, . . . , n} and suppose that A and B are, independently, equally likely to be any of the 2n subsets (including the null set and S itself) of S.

(a) Show that

P{A 鈯�B} =34n

Hint: Let N(B) denote the number of elements in B. Use

P{A 鈯�B} =鈭�i=0nP{A (鈯�B|N(B) = i}P{N(B) = i}

Show that P{AB = 脴} =34n

Two cards are randomly chosen without replacement from an ordinary deck of52 cards. Let B be the event that both cards are aces, let Asbe the event that the ace of spades is chosen, and letA be the event that at least one ace is chosen. Find

(a)role="math" localid="1647789007426" P(B|As)

(b) P(B|A)
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