Probability with and without replacement

  • Thread starter mutzy188
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In summary, the problem involves an urn with five balls, one marked WIN and four marked LOSE, where two players take turns selecting a ball. The first person to select the WIN ball wins. The probability of winning is 5/9 with replacement and 3/5 without replacement. To solve the problem, the probability of "L" when there are x balls in the urn is considered, and the formula P(W|x) = 1 - P(L|x) is used.
  • #1
mutzy188
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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

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
 
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  • #2
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...
 
  • #3
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?
 
  • #4
EnumaElish said:
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
 
  • #5
mutzy188 said:
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...
 
  • #6
mutzy188, do you get 3/5 for b?
 
  • #7
EnumaElish said:
mutzy188, do you get 3/5 for b?

Yes I did. Thank you very much for your help:smile:
 

1. What is the difference between probability with and without replacement?

Probability with replacement refers to an event where the selected item is put back into the group, while probability without replacement means that the selected item is not put back in the group.

2. How does probability with replacement affect the overall probability?

When an item is replaced, the probability of selecting that same item again remains unchanged. This means that each selection is independent, and the overall probability remains the same.

3. Does the order of selection matter in probability with and without replacement?

Yes, the order of selection can affect the overall probability. For example, if you are selecting two items without replacement, the probability of selecting a certain item first will affect the probability of selecting a different item second.

4. What is the formula for calculating probability with and without replacement?

The formula for probability with replacement is: P(A) = n/N, where n is the number of desired outcomes and N is the total number of outcomes. For probability without replacement, the formula is: P(A) = n/N x (n-1)/(N-1) x ... x (n-k+1)/(N-k+1), where k is the number of selections made.

5. How is probability with and without replacement used in real life situations?

Probability with and without replacement is used in various fields, such as in market research to determine the likelihood of a product being successful, in genetics to calculate the probability of certain traits being inherited, and in gambling to calculate the odds of winning a game. It is also used in everyday decision making, such as when choosing a random sample for a survey or when playing a game of cards.

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