If there are 7 doors in a room how many ways can a person enter one door and leave by another door
The GRE maths sample question given below is a problem solving question in permutation combination. This GRE sample question is an easy question. Show
Question 1: There are 5 doors to a lecture room. In how many ways can a student enter the room through a door and leave the room by a different door?
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You Pick | Prize Door | Don’t Switch | Switch |
1 | 1 | Win | Lose |
1 | 2 | Lose | Win |
1 | 3 | Lose | Win |
2 | 1 | Lose | Win |
2 | 2 | Win | Lose |
2 | 3 | Lose | Win |
3 | 1 | Lose | Win |
3 | 2 | Lose | Win |
3 | 3 | Win | Lose |
3 Wins (33%) | 6 Wins (66%) |
Here’s how you read the table of outcomes for the Monty Hall problem. Each row shows a different combination of initial door choice, where the prize is located, and the outcomes for when you “Don’t Switch” and “Switch.” Keep in mind that if your initial choice is incorrect, Monty will open the remaining door that does not have the prize.
The first row shows the scenario where you pick door 1 initially and the prize is behind door 1. Because neither closed door has the prize, Monty is free to open either and the result is the same. For this scenario, if you switch you lose; or, if you stick with your original choice, you win.
For the second row, you pick door 1 and the prize is behind door 2. Monty can only open door 3 because otherwise he reveals the prize behind door 2. If you switch from door 1 to door 2, you win. If you stay with door 1, you lose.
The table shows all of the potential situations. We just need to count up the number of wins for each door strategy. The final row shows the total wins and it confirms that you win twice as often when you take up Monty on his offer to switch doors.
Why the Monty Hall Solution Hurts Your Brain
I hope this empirical illustration convinces you that the probability of winning doubles when you switch doors. The tough part is to understand why this happens!
To understand the solution, you first need to understand why your brain is screaming the incorrect solution that it is 50/50. Our brains are using incorrect statistical assumptions for this problem and that’s why we can’t trust our answer.
Typically, we think of probabilities for independent, random events. Flipping a coin is a good example. The probability of a heads is 0.5 and we obtain that simply by dividing the specific outcome by the total number of outcomes. That’s why it feels so right that the final two doors each have a probability of 0.5.
However, for this method to produce the correct answer, the process you are studying must be random and have probabilities that do not change. Unfortunately, the Monty Hall problem does not satisfy either requirement.
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How the Monty Hall Problem Violates the Randomness Assumption
The only random portion of the process is your first choice. When you pick one of the three doors, you truly have a 0.33 probability of picking the correct door. The “Don’t Switch” column in the table verifies this by showing you’ll win 33% of the time if you stick with your initial random choice.
The process stops being random when Monty Hall uses his insider knowledge about the prize’s location. It’s easiest to understand if you think about it from Monty’s point-of-view. When it’s time for him to open a door, there are two doors he can open. If he chose the door using a random process, he’d do something like flip a coin.
However, Monty is constrained because he doesn’t want to reveal the prize. Monty very carefully opens only a door that does not contain the prize. The end result is that the door he doesn’t show you, and lets you switch to, has a higher probability of containing the prize. That’s how the process is neither random nor has constant probabilities.
Here’s how it works.
The probability that your initial door choice is wrong is 0.66. The following sequence is totally deterministic when you choose the wrong door. Therefore, it happens 66% of the time:
- You pick the incorrect door by random chance. The prize is behind one of the other two doors.
- Monty knows the prize location. He opens the only door available to him that does not have the prize.
- By the process of elimination, the prize must be behind the door that he does not open.
Because this process occurs 66% of the time and because it always ends with the prize behind the door that Monty allows you to switch to, the “Switch To” door must have the prize 66% of the time. That matches the table!
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If Your Assumptions Aren’t Correct, You Can’t Trust the Results
The solution to Monty Hall problem seems weird because our mental assumptions for solving the problem do not match the actual process. Our mental assumptions were based on independent, random events. However, Monty knows the prize location and uses this knowledge to affect the outcomes in a non-random fashion. Once you understand how Monty uses his knowledge to pick a door, the results make sense.
Ensuring that your assumptions are correct is a common task in statistical analyses. If you don’t meet the required assumptions, you can’t trust the results. This includes things like checking the residual plots in regression analysis, assessing the distribution of your data, and even how you collected your data.
For more on this problem, read my follow up post: Revisiting the Monty Hall Problem with Hypothesis Testing.
As for the Monty Hall problem, don’t fret, even expert mathematicians fell victim to this statistical illusion! Learn more about the Fundamentals of Probabilities.
To learn about another probability puzzler, read my post about answering the birthday problem in statistics!