by jason » Tue Feb 21, 2017 12:42 pm
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