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?
Bag A has marbles bbbw, bag B has bwww.
The odds of the 1st black marble being drawn from bag A was 3:4 (3 ways of drawing a black marble from A, 4 ways of drawing a black marble from the whole set of A+B).
The odds of drawing a second black marble from bag A are 3:4 times 3:4, or 9:16.
The odds of drawing a second black marble from bag B are 1:4 times 1:4, or 1:16.
Odds of drawing a second black marble from either bag are the sum of the odds for the two bags, or 10:16, which can be reduced to 5:8. Yup.
Figuring out loud.
