The probability of the prize being behind each door is .5, .3, .2 respectively. The participant opens door one. But the problem states that the fact that the contestant does not choose the original door is given. So this is a conditional probability. But I can't seem to figure out what the probability of changing the doors are.
If it is given that the contestant changes doors it means they would for sure switch to door two or door three right? So shouldn't the probability be 0.3+0.2=0.5? Because it is given that he would change the doors. But what would the probability of changing the doors be?
Any tips would help. Thank you guys so much!!!
You pick Door 1. The prize is behind that door 50% of the time. This means the prize is behind either door 2 or door 3 50% of the time, as well. So, once the announcer reveals one of the doors that does not contain the prize, the remaining door also has a 50% chance of being the correct door.
So, whether the contestant sticks with door 1 or changes to the remaining door, the probability of winning the prize (given the original probabilities) is 50%.
The contestant's best chance of winning is, start with Door 3, then allow the announcer to show a door, then switch to the remaining door. This gives the contestant an 80% chance of winning the prize.
Does this answer your question? I was not entirely clear what you were asking, and I can go into more detail if it would help.