On a TV game show, the contestant is asked
to pick one of three doors. Behind each door
are two large boxes to choose from. The
grand prize is in one of the boxes behind
Door 1. There are good prizes in one box
behind Door 2 and one box behind Door 3.
The other boxes all contain gag prizes.
What is the probability of winning the
grand prize?
d) What is the probability of winning a
good prize?
e) What is the probability of winning a gag
prize?
f) What is the sum of all the probabilities?
well, one box out of six = 1/ 6
or
right door = 1/3 then right box = 1/2 and indeed 1/3*1/2 = 1/6 anyway.
two out of six = 1/3
or
2/3 for one of the other doors then 1/2 for the right box = 1/3
If there is to be some outcome, any old outcome, the sum is one :)
To improve the chances of winning the grand prize, the contestant should follow the Monty Hall problem strategy. Here's how it works:
1. Initially, the contestant selects one of the three doors. Let's say they choose Door 1.
2. After the first choice, the host, who knows what's behind each door, opens one of the remaining doors to reveal a gag prize. This step is crucial and affects the probability.
3. At this point, there are two unopened doors: the one the contestant originally chose (Door 1), and one other door (either Door 2 or Door 3).
4. The contestant now has a choice to make: stick with their original choice (Door 1), or switch to the remaining unopened door (Door 2 or Door 3).
5. Counterintuitively, it is actually more advantageous to switch doors. The reason for this is that when the host reveals a gag prize behind one of the remaining doors, they are providing the contestant with extra information.
6. By switching doors, the contestant effectively doubles their chances of winning the grand prize. Since the host always reveals a gag prize, the probability of the grand prize being behind the door the contestant initially chose was 1/3. Therefore, the probability of the grand prize being behind one of the other two doors is now 2/3.
Therefore, to improve the chances of winning, the contestant should always switch doors after the host reveals a gag prize.