The paradox of the Monty Hall Paradox, or why being willing to revisit a decision can be beneficial in the long run. letsmakeadeal/ In a 70s game-show named Lets Make A Deal, the host, Monty Hall, would present three doors/curtains/boxes to a contestant and allow them to pick one of the boxes. After making a selection, Monty Hall would then display what was hidden in one of the other selections, showing it WAS NOT the big prize. The contestant was asked if they wanted to keep their selection or switch to the remaining door/curtain/box. The paradox is: Is it better odds for the contestant to keep their original selection, or switch to the remaining one? A lot of well-intentioned folks have argued that it is always better to switch because it increases the odds of winning from 1-in-3 to 2-in-3. math.ucsd.edu/~crypto/Monty/montybg.html It _IS_ true that being willing to switch gives you better odds, but the increase is from 1-in-3 to 1-in-2, or 50%. How can that be, you may ask? When you first chose, you have a 1-in-3 chance of winning. One of the three choices will be the winner, two will not be. Even if you chose correctly, Monty Hall presents you with another choice. If you chose wrong he will NEVER show you the grand prize, he always shows the other losing choice, but you dont know whether you picked the win or not. If you picked correctly (1-in-3), switching guarantees you lose. If you picked wrong, switching guarantees you win. If you enter the game saying Ill never change my choice, you have a 1-in-3 chance of winning each time you play. If you enter the game saying Ill always switch, you have a 1-in-2 chance of winning. If you enter the game saying Ill flip a coin for each keep-or-switch decision, you also have a 1-in-2 chance of winning. You have the following table which splits up the game possibilities into three groups based on a car in one of the three choices of A, B, or C. In the table, K means Keep, S means Switch, W means Win, and L means Lose. Look through the table and I think you will see it captures all possible scenarios for the game. Monty Hall Paradox Paradox ========================== Car is in A, Pick A, show B, K, W Car is in A, Pick A, show B, S, L Car is in A, Pick A, show C, K, W Car is in A, Pick A, show C, S, L Car is in A, Pick B, show C, K, L Car is in A, Pick B, show C, S, W Car is in A, Pick C, show B, K, L Car is in A, Pick C, show B, S, W Car is in B, Pick A, show C, K, L Car is in B, Pick A, show C, S, W Car is in B, Pick B, show A, K, W Car is in B, Pick B, show A, S, L Car is in B, Pick B, show C, K, W Car is in B, Pick B, show C, S, L Car is in B, Pick C, show A, K, L Car is in B, Pick C, show A, S, W Car is in C, Pick A, show B, K, L Car is in C, Pick A, show B, S, W Car is in C, Pick B, show A, K, L Car is in C, Pick B, show A, S, W Car is in C, Pick C, show B, K, W Car is in C, Pick C, show B, S, L Car is in C, Pick C, show A, K, W Car is in C, Pick C, show A, S, L Now, If you pick A, you have four rows from the first group and two rows each from the second two groups, a total of 8 scenarios. These are simply a rearrangement of the above groups based on pick first. Pick A, Car in A, show B, K, W Pick A, Car in A, show B, S, L Pick A, Car in A, show C, K, W Pick A, Car in A, show C, S, L Pick A, Car in B, show C, K, L Pick A, Car in B, show C, S, W Pick A, Car in C, show B, K, L Pick A, Car in C, show B, S, W There are eight possibles and half of them are winners. This holds for every combination of what you pick versus where the winner is located. 50-50. But if you decide ahead of time to always keep, the show and keep-vs-switch entries go away and you are left with: Pick A, Car in A, W Pick A, Car in B, L Pick A, Car in C, L Pick B, Car in A, L Pick B, Car in B, W Pick B, Car in C, L Pick C, Car in A, L Pick C, Car in B, L Pick C, Car in C, W Three W, six L, so 1-in-3 odds of winning. Being willing to decide again based on new information has increased your odds of winning from 1-in-3 to 1-in-2. Like anything else in life, FACTUAL INFORMATION IS YOUR FRIEND. Now, where is the paradox in Monty Halls Paradox? Because most folks, like the ucsd.edu link above, tell you that you should ALWAYS switch and present it as if that means you will win 2-out-of-3 times that you play. They even have a cute little live game which lets you pick a door and then walks you through the game, displaying live statistics based on Switched-vs-DID-NOT-SWITCH. The statistics make it look like you have 2-in-3 odds of winning if you play the game and ALWAYS SWITCH. But that is not the real situation, as demonstrated above. In fact, the statistics at ucsd actually show that the situation is 50-50 if you add up the total number of players and divide total winners by total players. It is about 50%. Switching doesnt give you 2-in-3 chances of winning. What switching does is allow the 2-in-3 original LOSING PICKS a chance at becoming winning picks. And _that_ is the paradox-within-a-paradox. Always be willing to review your decisions and where you are headed. It can turn LOSING choices in life into winning ones.
Posted on: Wed, 26 Mar 2014 13:36:36 +0000
Trending Topics
Recently Viewed Topics
© 2015