Probability with and without replacement

mutzy188

1. Homework Statement

An urn contains five balls, one marked WIN and four marked LOSE. You and another player take turns selecting a ball from the urn, one at a time. The first person to select the WIN ball is the winner. If you draw first, find the probability that you will win if the sampling is done

(a) with replacement - answer = 5/9
(b) without replacement - answer = 3/5

3. The Attempt at a Solution

(a) P(W) + P(LW) + P(LLW) + . . . . .

(1/5) + (4/5)(1/5) + (4/5)(4/5)(1/5) . . . .

so its is a series

SUM (1/5)(4/5)^(n-1) starting at n=1 to infinity
when I did this I got 1 and not 5/9

(b) P(W) + P(LW) + P(LLW) . . .

(1/5) + (4/5)(1/4) + (4/5)(3/4)(1/3) . . ..

SUM (1/5)( . . . and I got stuck here

Any Help would be appreciated

Thanks

Related Calculus and Beyond Homework Help News on Phys.org

learningphysics

Homework Helper
You have not considered that the second person is also taking out a ball... you did the problem as if it was just one person...

EnumaElish

Homework Helper
For me (b) is easier: Let P(L|x) be the probability of "L" when there are x balls in the urn. P(W|x) = 1 - P(L|x).

P(W|5) + P(L|5)P(L|4)P(W|3) + the 3rd term = 1/5 + 4/5 3/4 1/3 + the 3rd term.

Can you guess what the 3rd term is?

mutzy188

For me (b) is easier: Let P(L|x) be the probability of "L" when there are x balls in the urn. P(W|x) = 1 - P(L|x).

P(W|5) + P(L|5)P(L|4)P(W|3) + the 3rd term = 1/5 + 4/5 3/4 1/3 + the 3rd term.

Can you guess what the 3rd term is?
The third term should be:

P(L|5)P(L|4)P(L|3)P(W|2) = 1/5

learningphysics

Homework Helper
The third term should be:

P(L|5)P(L|4)P(L|3)P(W|2) = 1/5
Although the third term does come out to 1/5... just wanted to point out it should be:

P(L|5)P(L|4)P(L|3)P(L|2)P(W|1) = 1/5

because the expression P(L|5)P(L|4)P(L|3)P(W|2) means the second guy wins not you...

EnumaElish

Homework Helper
mutzy188, do you get 3/5 for b?

mutzy188

mutzy188, do you get 3/5 for b?
Yes I did. Thank you very much for your help Physics Forums Values

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