# Urn with marbles

Probability theory and statistics

### Urn with marbles

An urn contains 10 black marbles and some white ones. We are going to draw one marble at random. Before the draw, one more marble is added in the urn. Then we draw a random marble and it is a black one. What is the probability that the extra marble was black?

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### Re: Urn with marbles

Assuming the initial number of white marbles is x:
Before the addition of the extra marble:
Probability to draw a black marble: P(b)=10/(x+10)
Probability to draw a white marble: P(w)=x/(x+10)
The extra marble can be either black or white with probability 1/2 each.

We consider the below 2 cases:
a) Extra marble was black:
Probability to draw a black marble: P(b|b)=(10+1)/(x+10+1)=11/(x+11)
Probability to draw a white marble: P(w|b)=x/(x+1+10)=x/(x+11)

b) Extra marble was white:
Probability to draw a black marble: P(b|w)=10/(x+1+10)=10/(x+11)
Probability to draw a white marble: P(w|w)=(x+1)/(x+1+10)=(x+1)/(x+11)

Therefore, the requested probability (that the added marble was black) is:
[11/(x+11)]*1/2/{[10/(x+11)*1/2]+[11/(x+11)*1/2]} = 11/(10+11) = 11/21

jason

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