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A Horse Racing

You are betting on a horse race. There are only three runners, all having very similar odds and there is every indication that each has an equal chance of winning. You have already chosen your horse by picking straws and you are about to place the bet. Just before you do so, one of the other horses is withdrawn because the owner knows that it has no chance of winning due to a minor injury.

The conundrum is, do you stick with your original choice or do you swap to the other horse that is still in the running?

You might find the following somewhat counterintuitive. You should most definitely swap horses and place your bet on the other one. By doing so you increase your chances of winning from one in three (1/3) to two in three (2/3). Most people will believe that it makes no difference whether you change your choice or stick with your original choice and that both will have a fifty percent chance of winning, but this is not the case.

This is a beautiful example of the famous Monty Hall betting Problem where contestants chose one of three doors in order to win a car if they chose correctly, the car is behind one of them. The host would open one of the un-chosen doors that did not hide a car and would give the contestant the opportunity to change his choice. The answer is that he should always change, though this caused a huge amount of controversy even amongst scientists and mathematicians.

Originally there was a 1/3 chance that your horse would win and a 1/3 chance that either of the other horses would win. Thus there was a 2/3 chance that one of the other horses would win. Just because one of the horses was withdrawn does not change the odds that there is a 2/3 chance that one of those two horses would win. You just now know which one it will be. Place your bets.

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