Question

In a contest, the star prize is found in one of
three boxes, of which two are empty, the participant must choose a
of them. Once you have made your choice, the contest driver will
shows an empty box of the two that were not chosen by the contestant.
The driver asks the contestant if he keeps his choice or
it changes to the box that you had not chosen and that has not been opened - it suits you
change, stay the same or there is no difference in your choice? explain your
reply.

Answer 1

When you first selected a box, the chance of you winning a prize is = 1/3, Therefore, the probability of the prize being in the other two boxes is 2/3

Now, the driver opens one of the two empty boxes. This implies that the third box, that you did not choose and the driver didn't open, has 2/3 chance of having the prize.

Consider this through conditional probability

C- random variable of the chosen box having the prize

O- random variable of opening a box

P(C)=1/3

P(C|O) = P(O|C)*P(C) / [P(O|C)*P(C) + P(O|C^c)*P(C^c)]= Probability of winning if you stick with your choice

P(O|C) = The probability that the driver will open an empty box given that you chose the box with the prize= 1/2

Why? Because if you selected the prize box then the other two are empty. So, the probability of choosing empty box=1

P(O|C^c)= The probability that the driver will open an empty box given that you chose an empty box=1

Why? If you open an empty box, then 1 box is still empty, and the driver will open that box.

P(C^c) = 2/3

P(O|C)*P(C) / [P(O|C)*P(C) + P(O|C^c)*P(C^c)] = 1 * 1/3 / [1/3+2/3] =1/3

So, if you change your choice, the chances of winning increase to 2/3