monte hall question

Question

Suppose you are a participant in a game show. You have a chance to win a motorbike. You are asked

to select one of the 500 doors to open; the motorbike is behind one of the 500 doors; the other remaining doors

are losers and have balloons behind them. Once you select a door, the host of the game show, who knows

exactly what is behind each of the door, randomly opens 480 of the other doors all at once that s/he for sure

knows are losing doors and have balloons behind them. Then s/he reoffers you – whether you would like to

switch to the other doors or keep your initial or original selection as before. Now in this case, you are going

to make decision based on probabilistic reasoning. Therefore, whenever you are reoffered by the host to do

the selections among the remaining unopened doors, what is the probability of winning for each of the

remaining unopened doors (including your original selection)? Do you want to make a switch based on the

probabilistic reasoning? If you are switching, which door in particular do you want select later and why does

it matter? Show your work and calculation with relevant explanation in detail.

Answer 1

Go to play the game with 19 of your friends.

You all decide on one door.

You alone choose the strategy of always sticking with the original door;

that gives you a chance of 1/500 = 0.2% of getting the prize.

In contrast, your friends adopt the strategy of always switching;

each chooses one of the remaining closed doors.

One of you 20 guys is guaranteed to get the prize.

The chances of your friends getting are 1 - 1/500.

The chances of any particular one of your friends getting are (1 - 1/500)/19 > 5.2%.

I would want to be one of your friends.