| The door problem continued |
| On the last page we saw how provided that the host was aware of which door the car was behind you should (providing the probability of the car being behind each door is equal) switch doors after the first door has been opened. This assumed that the host knew what door the prize was behind what if he didn't? |
| We would return to the original result that I dimissed earlier. It will make no difference to your chance of winning if you switch doors or not. Your overall chance of winning however won't be the 1/2 that is assumed at the beginning, it will be 1/3. |
| Firstly let's look at the difference between this situation and the one proposed originally. In the original problem it is stated that the host opens a door to reveal a dustbin before he gives you the opporunity to change doors. In the new situation this would happen 2/3 of the time but 1/3 of the time the car would be revealed and you would loose there and then. (There are 3 doors so the chance the car is behind any chosen door is 1/3.) |
| This time there are 6 equally likely posibilities over what happens. These are listed below. (Once again you choose door 3) |
| Door number: 1 2 3 Door opened Possibility 1: C B B 1 Possibility 2: C B B 2 Possibility 3: B C B 1 Possibility 4: B C B 2 Possibility 5: B B C 1 Possibility 6: B B C 2 |
| Situation 1/ In possibilty 1 and possibility 4 you loose straight away without having the opporunity to change doors. The total probability of this is 1/6 + 1/6 = 1/3 (the 1/6 comes from the 6 equally likely outcomes) |
| Situation 2/ In possibility 5 and possibility 6 you win without switching doors. The total probability of this is 1/3. |
| Situtation 3/ In possibility 2 and possibility 3 you win by switching doors. The total proability of this is 1/3. |
| In the problem one door has already been opened and the prize hasn't been revealed therefore we can conclude that we are either in situation 2 or situation 3. There are two outcomes both of equal probability (each had an original probability of 1/3) and we now know that it has to been one of these two outcomes. We can, therefore, conclude that after the first stage of the original door being opened, provided the host doesn't know which door the car is behind, it is equally likely that the car is behind your chosen door and the other remaining door. |
| This is not saying though, that the chance of you finding the car is 1/2. We still have to get through the original stage of the first door being opened. The chance that the car ISN'T revealed at this stage is 2/3 (since the chance of it being behind the opened door is 1/3). Therefore the total probability of finding the car is 2/3 x 1/2 for each option. Whether you change doors or not, the overal chance of finding the car before the first door is opened is 1/3. |
| But how do I know which situation I am in? |
| How do you know if the host knows which door the car is behind? The answer to this question is easy. Watch the show a few times before you go on. If you are the first player of the game, tough luck you won't know (unless you ask the host). I will however for this example assume the show has been running for a fairly long time and you have/or can watch a number of shows before you play. |
| As I mentioned above, if the host doesn't know which door the car is behind you would expect the car to be revealed once in every three shows. So you watch the show 3 times, the car isn't revelaed that means the host knows which door the car is behind, correct? No, the nature of proabilities means it isn't that exact. If you throw a dice 6 times you wouldn't expect to get one 1, one 2, one 3, one 4, one 5 and one 6. |
| However once you have watched 9 shows you would have expected the car to appear (when the first door is opened) at least once. The probability that the car hasn't appeard when the first door is opened in 9 shows is 2/3 x 2/3 x 2/3 x 2/3 x 2/3 x 2/3 x 2/3 x 2/3 x 2/3 or (2/3)^9 (two thrids to the power of 9 (in normal notation the 9 will be in little superscript above the 2/3)). This equates to 0.026 or roughly 1 in 38 chance of this occuring. (That means if you watch 9 different shows 38 times you would expect to find that the car isn't revelaed in any of the 9 shows in roughly 1 of these 38 watches. (Of course if you watch the same 9 shows 38 times this doesn't count)). By the time you are watching say, 15 shows the chance that the car isn't revealed in any of these shows is (2/3)^15 or roughly 0.0023. |
| So if you have watched the show numerous time and never seen the car revealed when the first door is opened then it is fair to assume that the host knows which door the car is behind. You can never be certain (unless you ask the host) but the more time you have seen the show without the car being revealed when the first door is opened the less likely the chance the first door is chosen at random. |