What is the probability that a ball now drawn from the first bag is white?
One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn at random from teh second bag and is placed unseen into the first bag.
First bag: WWWWBBB
Second bag: WWWBBBBB
Now, before switching any balls, the probability of drawing a white ball from the first bag is 4/7. Since there are more black balls than white balls in the 2nd bag, transfering a ball from the 2nd bag to the 1st bag should make the 4/7 slightly lower, right? Just something to think about before starting.
Let's take each case separately:
Case 1: Transfered ball is black. Now the first bag would have WWWWBBBB, so probability of drawing white is 50%.
Case 2: Transfered ball is white. Now the first bag would have WWWWWBBB, so probability of drawing white is 5/8.
But what is the probability of case 1? The probability of drawing a black ball from the 2nd bag. (5/8)
Similarly, the probability of case 2 is 3/8.
We multiply the probabilities for each case, since the events are independent:
P(transfereing a black ball AND picking a white ball): `5/8*1/2` =5/16
P(transfering a white ball AND picking a white ball): `3/8*5/8` =15/64
Since either option is acceptable, we add their probabilities: