Because the solution to this riddle / puzzle is sometimes hard to see, or agree with, we will be presenting two different ways to explain it. If you're having a difficult time with the first way, please be sure to read the second explanation, it may help to make it clearer.
First, the answer....you should switch doors, it improves your chances.
The probability of selecting the door with the car is 1/3, and the probability of selecting a gag prize is then 2/3. Switching has increased the probability of winning the car to 2/3 from 1/3. Why?
The probability of selecting the right door is 1/3. The probability of losing is 2/3. You have more chance to lose than you have to win. You selected door 1, which has a probability of 1/3 of having the car. You have a 2/3 chance that you do not have the car. When the host reveals a gag prize behind one of the other doors, he has essentially just transferred 1/3 of the probability to you if you make the switch, giving you the 2/3 shot of getting the car. How?
Well change the test into the door you select and the other two doors. The probability that the car is in one of those two doors is 2/3. You now know that one door does not have the car. By switching, you now have the 2/3 shot of getting the car because you now have one of the doors in the group with the 2/3 probability. Still don't see it, here's another way to look at it.
Let's say that you tried this game picking one of three doors 300 times. On average, you would pick the correct door only 33% of the time or 100 times out of the 300 attempts. This means that you would be wrong 200 times.
If you only have a 33% chance, then there is a 66% chance that one of the other 2 doors has the car. Once the host eliminates one of the other two doors, you now own that 66% chance. If you switch, you will win, on average, 200 times
Therefore, out of 300 trials, even if you were to switch every time, you would lose the car 100 times but, you would win 200 times. Which odds are better, to win 100 times out of 300 or 200 times out of 300?
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