You choose one of two identical looking bags at random. One bag has three black marbles and one white marble. The other has three white marbles and one black marble.

After choosing a bag you draw one marble out at random. You notice it is black. You then put it back and draw another marble out of the same bag at random.

What is the probability that the second marble drawn is black?

If there were a million-and-one marbles in each bag (1m black only 1 white and vice versa) and your first draw was black you would be pretty convinced (probability 1000000/1000001) you'd picked the black-heavy bag and the next one out was probably going to be black. Since these bags are also unevenly filled, the colour of your first draw gives the best indication of the colour of the majority of the bag (which indicates the final probability will be higher than 1/2)
Here the fact your first marble is black gives you a 3/4 probability you have the bag with three blacks and 3/4 probability the next one will be.
3/4*3/4 = 9/16
You could, however, have got 'lucky' and picked the first black marble from a white-heavy bag. It's less likely, 1/4, but it could be the case. If so, your chances of picking the lone black marble again are 1/4.
1/4*1/4 = 1/16
The total probability is therefore 9/16 + 1/16 =10/16
=5/8

Posted by Lee on 2003-10-14 01:40:40