Question

Monty Hall Problem: How was probability used in arriving at the answer? What probability ideas does this demonstrate and use? Explain and give examples. You may use other sources as well but make sure to cite them.

Answer 1

In the monty hall problem, the probability concept is used is as follows:

There are 3 doors, behind two of which is a goat and behind one of them is a winning prize.

Now initially suppose We select door 1.

Now there are 3 possibilities here:

Choosen Door	Prize door	Door opened	Price won after changing door
1	1	2 or 3	No
1	2	3	Yes
1	3	2	Yes

Note that each of the door above is equally likely to have the prize intially, therefore all the above 3 cases are equally likely. We see that if the person selects door 1, and suppose door 3 has the prize, then the game host would show him the door 2 and offer him to change his choice or stick to door 1. Similarly would be the case when the price is behind door 2. He would only lose when the prize is behind door 1 itself.

Therefore out of 3 cases, the person would win the prize in 2 cases when he switches his door after the game host opens up one of the door after his choice. Therefore 2/3 is the probability of winning and therefore the choice should be switched after one of the door is opened.