### The Monty Hall Problem

Suppose you’re on a game show, and 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 [but the door is not opened], 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?

Ah! Click here to read the solution.

The solution presented by Vos Savant shows the three possible arrangements of one car and two goats behind three doors and the result of staying or switching after initially picking Door 1 in each case:

behind Door 1 behind Door 2 behind Door 3 result if staying at door #1      result if switching to the door offered
Car Goat Goat Car     Goat
Goat Car Goat Goat     Car
Goat Goat Car Goat     Car

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. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.

### Peter The Pirate

Pirate Peter had been captured by a Spanish general and sentenced to death by his 50-man firing squad.

Peter cringed, as he knew their reputation for being the worst firing squad in the Spanish military. They were such bad shots that they would often all miss their targets and simply maim their victims, leaving them to bleed to death, as the general’s tradition was to only allow one shot per man to save on ammunition. The thought of a slow painful death made Peter beg for mercy.

“Very well, I have some compassion. You may choose where the men stand when they shoot you and I will add 50 extra men to the squad to ensure someone will at least hit you. Perhaps if they stand closer they will kill you quicker, if you’re lucky,” snickered the general. “Oh, and just so you don’t get any funny ideas, they can’t stand more than 20 ft away, they must be facing you, and you must remain tied to the post in the middle of the yard. And to show I’m not totally heartless, if you aren’t dead by sundown I’ll release you so you can die peacefully outside the compound.

I must go now but will return tomorrow and see to it that you are buried in a nice spot, though with 100 men, I doubt there will be much left of you to bury.”

After giving his instructions the general left. Upon his return the next day, he found that Peter had been set free alive and well. “How could this be?” demanded the general. “It was where Peter had us stand,” explained the captain of the squad.

Where did Peter tell them to stand?

Ah! Click here to read the solution.

Peter told them to form a circle around him. All the squad was facing in at Peter, ready to shoot, when they realized that everyone who missed would likely end up shooting another squad member. So no one dared to fire, knowing the risk. Thus at sundown he was released.