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April 10th, 2018, 05:16 AM   #1
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Conditional probability

I hope someone can help!

My problem is an analogy of a work question and I can’t solve it for the life of me 🙈

• You have a bag of 10 marbles
• 8 are black
• 2 are white
• You select a marble
• If it is black, you put the marble back and pick again
• If it is white you stop selecting
• After a maximum of 4 selections, what is the probability that you will have selected a white?

Any help would be very much appreciated!!
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April 10th, 2018, 10:17 AM   #2
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let $P[W|k]$ be the probability of picking white in $k$ picks

note it's assumed picks are independent of one another.

$P[W]=P[W|1 ]+P[W|2 ]+P[W|3 ]+P[W|4] =

\dfrac 1 5 + \left(\dfrac 4 5\right)\left(\dfrac 1 5\right)+\left(\dfrac 4 5\right)^2\left(\dfrac 1 5\right)+\left(\dfrac 4 5\right)^3\left(\dfrac 1 5\right)=\dfrac{369}{625}$
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April 10th, 2018, 01:30 PM   #3
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It is easy to compute the probability of P, a black being selected each time, since you are putting it back each time. P$=(\frac{4}{5})^4$. So the probability that a white had been selected is 1-P.
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