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Chapter 3: Q.3.24 (page 111)

Rank the following from most likely to least likely to occur:

1. A fair coin lands on heads.

2. Three independent trials, each of which is a success with probability.8, all result in successes.

3. Seven independent trials, each of which is a success with probability 9, all result in successes.

Short Answer

Expert verified

Three independent trails are ranked first, second, and third0.4783<0.5<0.512P3<P1<P2Use the definition of independence in conjunction with the probability of intersections

Step by step solution

01

Step1:A fair coin lands on heads. (part a)

P1=P(fair coin lands on head=1/2

02

Step2:Three independent trails(part b)

independence

E1,E2,鈥�,Enare independent鈬�PE1E2鈥�En=鈭�i=1nPEi

P(one trial ends in success )=0.8

Therefore

P2=P(3Independent trials are successful)=P One trial is successful. ^3=0.8^3=0.512

03

Step3:Seven independent trail(part c)

E1,E2,鈥�,Enare independent鈬�PE1E2鈥�En=鈭�i=1n鈥�PEi

Given

P( one trial ends in success )=0.9

Therefore

P3=P(7Independent trials are successful. One trial is successful.)^7=0.97鈮�0.4783

0.4783<0.5<0.512P3<P1<P2

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Most popular questions from this chapter

In Laplace鈥檚 rule of succession (Example 5e), suppose that the first nflips resulted in r heads and n鈭�rtails. Show that the probability that the(n+1)flip turns up heads is (r+1)/(n+2). To do so, you will have to prove and use the identity

鈭�01yn(1-y)mdy=n!m!(n+m+1)!

Hint: To prove the identity, let C(n,m)=鈭�01yn(1-y)mdy. Integrating by parts yields

C(n,m)=mn+1C(n+1,m-1)

Starting with C(n,0)=1/(n+1), prove the identity by induction on m.

(a) A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin?

(b) Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin?

(c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

Suppose we have 10 coins such that if the ith coin is flipped, heads will appear with probability i/10, i = 1, 2, ..., 10. When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the fifth coin

A total of 48 percent of the women and 37 percent of the men who took a certain 鈥渜uit smoking鈥� class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class was male,

(a) what percentage of those attending the party were women?

(b) what percentage of the original class attended the party?

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?
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