The Monty Hall “Paradox” isn’t actually logical paradox. It’s counterintuitive probability.
Game shows and casinos deliberately rig the probabilities to ensure the house wins. Sometimes, they rely on ignorance as did the gameshow Let’s Make a Deal.
The game went like this. Behind closed doors, there are two goats and one car. They let the player pick on of the doors.
The host of the show then opens one of the doors to reveal a goat (he always reveals a goat). The host asks the player if he wants to switch doors or stay with his original guess.
Would you Stay or Switch?
This is where cognitive bias kicks in. Many people on the game show stuck with their original choice even though this was irrational.
Think about it this way. On the first turn, you have a 1/3 chance of picking the door with the car and a 2/3 chance of picking a goat. This is obvious so far.
After the host opens the door and shows the goat, people make a mistake calculating their odds. They believe they have two choices – a goat or a car, with a 1/2 chance of each. This is what the host wants you to think.
There are still three doors and the results of the first turn must be considered. The past cannot be ignored in the second turn. You update your past estimate based on new data.
Remember, you had a 2/3 chance of picking the wrong door in the first turn. There are still three doors, but one option is now closed. You have two choices between three doors. So update your probability for the second turn.
If you stay, you have a 1/3 chance of picking the car and a 2/3 chance of picking the goat.
If you switch, you have a 2/3 chance of picking a car and 1/3 chance of picking the goat.
If that isn’t clear yet, think about it.
Every so often MIT nerds go to Las Vegas and make a fortune off of card games like Blackjack. The games are not rigged, like slot machines, so a good mind can count the cards and update his probabilities and beat the house.
Isn’t Probability fun? Too often, we fail to properly update our probability estimates. You can’t ignore the past and you can’t select which data to pay attention to. This is a superficial example, sure, but this shows a real problem with the way we are used to thinking.