Question

Sometimes probability statements are expressed in terms of odds. The odds in favor of an event A are the following ratio.

P(A)/P(not A) = P(A)/P(Ac)

For instance, if P(A) = 0.60, then P(Ac) = 0.40 and the odds in favor of A are 0.60/ 0.40 = 6/4 = 3/2 , written as 3 to 2 or 3:2.

(a) Show that if we are given the odds in favor of event A as n:m, the probability of event A is given by the following. P(A) = n/n + m Hint: Solve the following equation for P(A). n/m = P(A)/1 − P(A)

n(1 − P(A)) = _____ (P(A))

n − nP(A) = ____P(A)

n = ____P(A) + nP(A)

n =_____ P(A)

n/n + m = P(A)

(b) A telemarketing supervisor tells a new worker that the odds of making a sale on a single call are 6 to 19. What is the probability of a successful call? (Round your answer to two decimal places.)

(c) A sports announcer says that the odds a basketball player will make a free throw shot are 3 to 5. What is the probability the player will make the shot? (Round your answer to two decimal places.)

When do creative people get their best ideas? USA Today did a survey of 966 inventors (who hold U.S. patents) and obtained the following information.

Time of Day When Best Ideas Occur
Time	Number of Inventors
6 A.M.-12 noon 12 noon-6 P.M. 6 P.M.-12 midnight 12 midnight-6 A.M.	292 131 328 215

(a) Assuming that the time interval includes the left limit and all the times up to but not including the right limit, estimate the probability that an inventor has a best idea during each time interval: from 6 A.M. to 12 noon, from 12 noon to 6 P.M., from 6 P.M. to 12 midnight, from 12 midnight to 6 A.M. (Enter your answers to 3 decimal places.)

6AM-12PM	12PM-6PM	6PM-12AM	12AM-6AM

John runs a computer software store. Yesterday he counted 123 people who walked by the store, 52 of whom came into the store. Of the 52, only 23 bought something in the store. (Round your answers to two decimal places.)

(a) Estimate the probability that a person who walks by the store will enter the store.


(b) Estimate the probability that a person who walks into the store will buy something.


(c) Estimate the probability that a person who walks by the store will come in and buy something.


(d) Estimate the probability that a person who comes into the store will buy nothing.

Answer 1

1.a.

The odds in favour of A is n:m

Hence,n*(1-P(A))=m*P(A)

or,n-n*P(A)=m*P(A)

or,n=(n+m)*P(A)

or,P(A)=n/(n+m)

b.

P[of a successfull call]=no of cases of a successfull call/(no of cases of a successfull call+no of cases of an unsuccessfull call)=6/(6+19)=6/25=0.24

c.

P[the player will make a shot]=no of cases where the player will make a shot/(no of cases where the player will make a shot+no of cases where the player will not make a shot)=3/(3+5)=3/8=0.38

2.Total no of inventors=966

Time	No of inventors	Probability
P[6AM-12PM]	292
P[12PM-6PM]	131
P[6PM-12AM]	328
P[12AM-6AM]	215

3.a.P[the person who will walk by the store will enter the store]=no of persons who enter the store/no of persons who walk by the store=52/123=0.42

b.P[the person who will walk by the store will buy something]=no of persons who buys something/no of persons who walk by the store=23/123=0.19

c.P[a person who walks buy the store will come in and buy something]=P[a person who walks buy the store will come in]*P[a person who come in will buy something]=(52/123)*(23/52)=23/123=0.19

d.P[a person comes in the store and buys nothing]=(52/123)*(29/52)=29/123=0.24