Hello, this is Dozie from DeeNaija. During one of my daily research, I found something interesting, The Monty Hall Problem.

Suppose you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Common-sense would say it doesn’t really matter whether you switch or stay with the door you picked earlier, as there is a 50%(1/2) chance that you would see a car if you stay with the door you earlier picked. Well, that is not entirely true. Before I explain further, I’d be listing the assumptions made for the game show:
♠ The host must always open a door that was not picked by the contestant.

♠ The host must always open a door to reveal a goat and never the car.

♠ The host must always offer the chance to switch between the originally chosen door and the remaining closed door.

According to Marilyn Vos Savant (an American who is known for having the highest recorded IQ according to the Guinness Book of Records), Under the standard assumptions,the contestant should switch, as the contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their initial choice have only a 1/3 chance.

A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three.

An intuitive explanation is that, if the contestant initially picks a goat (2 of 3 doors), the contestant will win the car by switching because the other goat can no longer be picked, whereas if the contestant initially picks the car (1 of 3 doors), the contestant will not win the car by switching. The fact that the host subsequently reveals a goat in one of the unchosen doors changes nothing about the initial probability.

Another way to understand the solution is to consider the two original unchosen doors together. As Cecil Adams puts it (Adams 1990), “Monty is saying in effect: you can keep your one door or you can have the other two doors.” The 2/3 chance of finding the car has not been changed by the opening of one of these doors because of Monty, knowing the location of the car, is certain to reveal a goat. So the player’s choice after the host opens a door is no different than if the host offered the player the option to switch from the originally chosen door to the set of both remaining doors. The switch in this case clearly gives the player a 2/3 probability of choosing the car.

As Keith Devlin says (Devlin 2003), “By opening his door, Monty is saying to the contestant ‘There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I’ll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.'”

Vos Savant suggests that the solution will be more intuitive with 1,000,000 doors rather than 3. (Vos Savant 1990a) In this case, there are 999,999 doors with goats behind them and one door with a prize. After the player picks a door, the host opens 999,998 of the remaining doors. On average, in 999,999 times out of 1,000,000, the remaining door will contain the prize. Intuitively, the player should ask how likely it is that, given a million doors, he or she managed to pick the right one initially. Stibel et al. (2008) proposed that working memory demand is taxed during the Monty Hall problem and that this forces people to “collapse” their choices into two equally probable options. They report that when the number of options is increased to more than 7 choices (7 doors), people tend to switch more often; however, most contestants still incorrectly judge the probability of success at 50:50.

View this Post with images and demonstration @DeeNaija: https://deenaija./2017/08/25/monty-hall-problem-when-common-sense-isnt-an-advantage/