Question

A contestant on a game show is offered a choice of 4 boxes. one of the boxes contains a diamond and the other three boxes each contain a pebble. tThe contestant cannot see into the boxes and so does not know which box contains the diamond. First the ontestant selexts a box. Host opens it ait to reveal a pebble. ( The host know the boxes and choose normaly between boxes containing pebble.). The host then ask the contestant if he want to change his election and after this host opens anothe box containing a pebble. the host ask the contestant if he wants to switch again. what is the probability that the contestant will win the diamond if he changes his selection both times?

Answer 1

Answer :

There are four boxes totally where box 1, box 2 and box 4 contains pebbles and box 3 contains diamond.

The contestant first selects box 2 containing pebble. So the host can select only box 1 or box 4 as he cannot open box 3 with diamond and the box 2 which the contestant has selected. If he stay with box 2, he cannot win. On switching to box 3 he can win diamond.

Thus if the contestant is right the first time, staying pays off ( which happens 1 out of 4 times).

And if the contestant is wrong the first time, switching pays off ( which happens 3 out of 4 times).

The following table summarises the result of STAYING:

Notice that the contestant can win in 4 of 16 equally likely situations. So the probability of winning diamond is 4 / 16 = 1 / 4  if the contestant stay with initial choice.

The following table summarises the result of SWITCHING:

Notice that the contestant can win in 12 out of 16 equally likely situations. So the probability of winning diamond is 12 / 16 = 3 / 4 if the contestant switch when he has chance. Thus if the contestant switch he will be more likely to win.

ABSTRACT REASONING:

   At the beginning of the game, the diamond is equally likely to be in box 1,2,3 or 4. Thus there is 1/4 chance that the diamond is in box 1, 1/4 chance that the diamond is in box 2, 1/4 chance that the diamond is in box 3,1/4 chance that the diamond is in box 4.

This means that there is 1/4 chance that the diamond is in the box that contestant selects and 3/4 chance that the diamond is in one of the other three boxes.

​Thus if the contestant choose to stay he will win 1/4 of the time; If he uses the switching strategy he will win 3/4 of the time, making the switching strategy better.

​The reason why the switching strategy is better is that by switching, the contestant will win whenever the diamond is in either of the boxes he did not initially choose.