I have several blog post ideas drafted at this point. Some dealing with culture and Christianity, but tackling topics such as those with the correct amount of tact requires a level of polish that takes me some time to achieve. In the mean time, I basically just resurrected an old Facebook Note I wrote about the Monty Hall problem and updated it a bit. Here goes:
The Monty Hall Problem is always a fun one. I've met several people who refuse to believe the answer no matter how I explain it.
The Problem:
The Monty Hall Problem originates from the game show Let's Make A Deal. The Contestant is presented with 3 doors and told that behind one of them is a car, but the other 2 are junk.
Here's how the game plays:
- The Contestant chooses a door.
- The Host opens up one of the remaining doors (they will ALWAYS open a door that has junk behind it. This is important.)
- With the choices now narrowed to 2, the Contestant is then asked if they want to stay with their original choice or switch to the remaining door.
The question is, is it better to switch or stay?
The Answer:
Most people believe that the odds are 50/50 at this point. In practice, most contestants will choose to stay. The oddity of this problem is that, in all actuality, the contestant has a better chance of winning the car if they SWITCH.
Why:
The reason for this probability oddity is in step 2 of the game. If the choice of doors in the second step was actually random, then there would be equal probabilities of 3 different outcomes:
- The Contestant's first choice is right and the Host opens a Junk Door
- The Contestant's first choice is wrong and the Host opens a Junk Door
- The Contestant's first choice is wrong an the Host opens the Car Door effectively cutting the game short at step 2 and making the final dilemma pointless
The point is, you'd still only have a 1/3 chance of winning if you STAY with your original door.
Because the Host only opens a Junk Door at step 2, the probabilities for choosing to SWITCH are altered. Now you have 2 choices, bet that you were right with your first guess, or bet that your first guess was wrong. Since there was a 2/3 chance you were wrong to begin with, choosing to SWITCH will result in winning 2/3 of the time.
Because the Host only opens a Junk Door at step 2, the probabilities for choosing to SWITCH are altered. Now you have 2 choices, bet that you were right with your first guess, or bet that your first guess was wrong. Since there was a 2/3 chance you were wrong to begin with, choosing to SWITCH will result in winning 2/3 of the time.
One mental exercise that can make this clearer is if you decide on switching or staying before you even pick a door.
If you decide that you are going to STAY in advance then you have to pick the Car Door in order to win. A 1 in 3 probability.
If you decide to SWITCH in advance then you just have to pick a junk door in order to win. That's a 2 in 3 probability.
If neither of these convince you, then I created a probability tree of the possible steps and outcomes:
