Question

The Monty Hall Game Show Problem: In a TV game show, a contestant selects one of three doors; behind one of the doors there is a prize, and behind the other two there are no prizes. After the contestant selects a door, the game-show host opens one of the remaining doors, and reveals that there is no prize behind it. The host then asks the contestant whether they want to SWITCH their choice to the other unopened door, or STICK to their original choice. Is it probabilistically advantageous for the contestant to SWITCH doors, or is the pro- bability of winning the prize the same whether they STICK or SWITCH? (assume that the host selects a door to open, from those that are available and do not have a prize behind them, with equal probability).

Answer 1

Answer:

is more advantageous to switch the door

Step-by-step explanation:

To solve this issue is convenient to divide in possible steps and scenarios:

1° Step) Choose a door

a) You can choose the door with the price → 2/3 of probability

b) You can choose the door that has not the price  → 1/3 of probability

2° Step) Monty shows you a door. Since he will not show you the door with the prize because you would switch to that door,  then

a) If you chose the door with the price (1/3 of probability)→ the 2 remaining doors do not have the price and Monty will show you one of them  → If you switch , you lose and if you stay you win.

b)  If you choose the door that has not the price (2/3 of probability) → there is only one door  that has not the price and Monty will show you that one  → If you switch, you will choose the door with the price and you win , if you stay you lose.

Therefore initially you had

-1/3 probability of wining

after he showed you the wrong door

- 2/3 of probability of winning if you switch (1/3 more)

is more advantageous to switch the door