# Probability of Winning Prize Behind 3 Doors: 30% or 70%?

• sapiental
I'm sorry! You are right and I am wrong!In summary, the probability of picking the correct door in the first situation, where the contestant always sticks with their first pick, is 1/3. In the second situation, where the contestant switches their pick after one false door has been shown, the probability of picking the correct door becomes 2/3. This has been well covered by Marilyn Vos-Savant and others.
sapiental
Hello,

for the two following situations give the ultimate probability.

There is a prize behind one of three doors.

1. Picking a door out of 3 doors and always sticking with your firrst pick.
2. Picking a door out of 3 doors and alwats SWITCHING your pick after one false door has been shown.

for 1 I get the probability is around 30% and for 2 I get around 70%.

Is this right? Thanks

Oh, dear, not again! Monte Hall should burn in hell!
Did you notice, by the way, that you never actually asked a question? I'm not at all sure what you mean by "probability is around 30% and for 2 I get around 70%"
I will ASSUME that you meant "what is the probability of picking the prize if (1) you do not switch after the false door has been opened, (2) you do switch after the false door has been opened.

If you are faced with three doors and have NO idea which door the prize is behind, then it is equally likely to be behind anyone and so ther probability of picking the correct one is 1/3. If you do not switch when the false door is opened then the opening is irrelevant and so the probability of picking the correct door is 1/3. I have no idea why you would say "around 30%".

Assuming that the person who opens the "false" door knew where the prize was and intentionally opened a door the prize was not behind, then then the prize is equally likely to be behind either of the two doors. Switching makes your probability 1/2, not 70%.

Some people feel that, since the probability, after the false door was opened, is equal for each remaining door, it doesn't matter whether you switch or not. "Marilyn Vos-Savant"s response to that was very good: Suppose there were 10000 doors with a prize behind exactly one. You pick one at random (so the chance that the door you picked has the prize is 1/1000). The M.C. then opens 9998 false doors leaving only yours and one other door the prize can be behind. Would you switch?

By the way, it is interesting to show that, if the M.C. does NOT know, himself, which door the prize is behind, but, by cnance, opens a door the prize is NOT behind, the probability of getting the prize, whether you switch or not, is 1/3.

HallsofIvy said:
Assuming that the person who opens the "false" door knew where the prize was and intentionally opened a door the prize was not behind, then then the prize is equally likely to be behind either of the two doors. Switching makes your probability 1/2, not 70%.

No, here the probability of getting the prize by following the Switching strategy becomes 2/3. That's pretty close to 70%, although I would prefer the exact value.

I thought this ground has been well covered by Marilyn (and others).

Once, I got my friend to agree to a five dollar game of this with a pack of cards. He picked a card, then I removed fifty cards that weren't the ace of spades from the deck. Since he was convinced the odds were even, I said if he had the ace of spades at the end, he got ten bucks, if I had it, I got five bucks.

After I removed the fifty cards he decided he didn't want to play :D

Assuming that the person who opens the "false" door knew where the prize was and intentionally opened a door the prize was not behind, then then the prize is equally likely to be behind either of the two doors. Switching makes your probability 1/2, not 70%.

im not even good at math, and i agree with marilyn.

there's three doors.. so you initially have a 2/3 chance of picking the wrong door:
goat, goat, prize

so when you switch doors you have a 2/3 chance of picking the right door.

makes sense to me.

~Amy

Okay, Okay!

## 1. What does the probability of winning a prize behind 3 doors mean?

The probability of winning a prize behind 3 doors refers to the likelihood of selecting the correct door that has a prize behind it, out of a total of three doors. In this scenario, there is a 30% chance that the prize is behind one door and a 70% chance that it is behind another door.

## 2. How is the probability of winning a prize behind 3 doors calculated?

The probability of winning a prize behind 3 doors is calculated by dividing the number of favorable outcomes (winning the prize) by the total number of possible outcomes. In this case, since there are 3 doors, the total number of possible outcomes is 3, and the number of favorable outcomes is 1 (since there is only 1 prize). This yields a probability of 1/3 or 0.33, which is equivalent to 30%.

## 3. Why is the probability of winning a prize behind 3 doors only 30%?

The probability of winning a prize behind 3 doors is only 30% because there are 3 possible outcomes, but only 1 of those outcomes results in a win (the other 2 outcomes result in a loss). This means that for every 3 attempts, only 1 will result in a win, giving a 30% chance of winning.

## 4. Can the probability of winning a prize behind 3 doors change?

Yes, the probability of winning a prize behind 3 doors can change depending on the scenario. If the number of doors or the number of prizes behind the doors is different, the probability will change. For example, if there are 6 doors and 2 prizes, the probability of winning will be 2/6 or 0.33, which is equivalent to 33%. Additionally, if the doors are not equally likely to have a prize behind them, the probability will also change.

## 5. How can the probability of winning a prize behind 3 doors be increased?

The probability of winning a prize behind 3 doors can be increased by reducing the number of doors or increasing the number of prizes behind the doors. For example, if there are only 2 doors and 1 prize, the probability will be 1/2 or 0.50, which is equivalent to 50%. Additionally, if the doors are equally likely to have a prize behind them, the probability can also be increased by selecting a door at random, rather than choosing a specific door.

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