There is a basket with 5 balls; 3 red and 2 white. If in the first pick, you choose a ball, you add another ball of the same color to the basket along with it (if you pick a red ball, now there will be 4 red and 2 white balls; similarly with the white). Now given that in the second pick, you draw a white ball, what is the probability of getting the red ball in the first pick?
Using the Bayes theorem, the answer would be 1/2. But, say if we didn't have the information for the second pick, the answer would have been 3/5.
This now is my question: why is there a change in the probabilities when there is new information given?
when you have picked a white ball at second pick then it is more likely to have picked a with ball at the first time that has increased the probability of picking the with ball in second time.
picking a red ball then white ball $$3/5 * 2/6=1/5$$ picking a white ball then white ball $$2/5 * 3/6=1/5$$ picking a red ball then red ball $$3/5 * 4/6=2/5$$ picking a white ball then red ball $$2/5 * 3/6=1/5$$ so what happened is one of the first two's and we want the probability of picking a red ball at first if we have picked a white one at second time $$ \frac{pr(red \space then \space white)}{pr(red\space then\space white) + pr(white\space then\space white)} = \frac{1/5}{1/5+1/5}$$