Denote by A B and C the events that a grand prize is behind

. Denote by A, B, and C the events that a grand prize is behind doors A, B, and C, respectively. Suppose you randomly pick a door, say A. The game host opened a door, say B, and showed there was no prize behind it. Now the host offers you the option of either staying at the door that you picked A or switching to the remaining unopened door C. Use probability to explain whether you should switch or not.

Solution

the probable choices are

So, the probability of winning if stayed on the choice is 1/3

where as   the probability of winning if switched is 2/3

So, one should switch

Door A	Door B	Door C	Choose A	Choose C
Prize	No	No	Won	Lost
No	Prize	No	Lost	Won
No	No	Prize	Lost	Won