### Problem 3 - Guess and make a fortune game

In a game of 'guess and make fortune', the rules are as follows.

1. There are three boxes and hidden under one of them is a huge amount of money. There is nothing under the rest two boxes

2. The host knows where exactly the fortune is

3. The player has to choose one of the three boxes

4. After the player chooses a box, the host opens one of the remaining 'empty' boxes

5. The player is now allowed to switch his choice of box if he wishes

The question is, does the player have a higher probability of winning if he makes a switch.

As always, the rule to solve simple everyday probability problems is to think simple. Afterall CAT tests your thinking skills more than your ability to do complicated math

Click here for the answer

1. There are three boxes and hidden under one of them is a huge amount of money. There is nothing under the rest two boxes

2. The host knows where exactly the fortune is

3. The player has to choose one of the three boxes

4. After the player chooses a box, the host opens one of the remaining 'empty' boxes

5. The player is now allowed to switch his choice of box if he wishes

The question is, does the player have a higher probability of winning if he makes a switch.

As always, the rule to solve simple everyday probability problems is to think simple. Afterall CAT tests your thinking skills more than your ability to do complicated math

Click here for the answer

## 7 Comments:

the probability remains the same, this is similar to the previous problems

The probability remains the same as even he who selected one box Knows this thing that there is box with no money so no tension for him and he would not switch ...

My comments will come by evening or may be I will give it an other day. I want people to think and give it a try.

You need to think on the subtle differences between the previous problem and this. That understanding will give you a stronger hold on probability.

And note, that I am not saying that your answer is wrong, but all I am saying is that this problem is not as simple as the previous one :)

I dint quite got the problem..

1) if the box that the host opens is the one where the player will get the prize if its there then the probability is zero as if the player selects the box with the money, the host will open the empty box and if the player selects the empty box, then the host will open the empty box which is remaining.

2) if the prize is the box which the players selects then the probability is 1/3.

sorry but maybei dint followed the question properly.

Cheers

Atul

Good to see the old timers :)

To make the problem easier to understand.

Say there are three boxes A, B and C. Say the player picks B. The host will open one among A and C which he knows is empty. Does it increase the probability to win if the player now switches is the question

The probability remains the same since for the player the situation is still pretty much the same .... he has to chose one box out of the three boxes which is empty , its the same as if the host opens the box shows that its empty and closes it , the player still has to find the box with gold from among the three boxes

The answer is that the contestant is better off making a switch.

It is very difficult to articulate the answer is such questions, but let me make a try. If people dont get it, then I will make more tries from different angles

Here goes the argument:

1. When the player chooses the box in the beginning of the game, the probability of him getting the fortune is 1/3.

2. Let us divide the boxes into two sets. One contains the box that the player has choosen and the second set contains the two boxes that the player has not choosen.

3. The probability that the fortune is in the second set is 2/3.

4. Now the host comes and removes an empty box from the second set. Now if the player is given a chance, he is better off choosing the second set because the probability of winning is 2/3

If you are still confused and half convinced, think that there were 100 boxes and fortune behind one. Now you choose one box and the host eliminates 98 empty boxes out of the rest 99. Wont you make a switch?

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