Welcome to the Math312 Game Show! Where Probability Helps You Win Stuff! The Gam
ID: 3354822 • Letter: W
Question
Welcome to the Math312 Game Show! Where Probability Helps You Win Stuff!
The Game:
Suppose you're on a game show, and you're given the choice of 3 doors. Behind one door is a car, behind the others, goats. You start by choosing a door, say number 1, which remains closed for now. The game show host, who knows what's behind the doors, opens another door, say number 3, which reveals a goat. He says to you, "You've already chosen door number 1, now that I've shown you a goat behind door number 3, do you want to switch to door number 2? Or stay with you original choice of door number 1?" After you choose to stay or switch doors, they are opened to see if you won!
The Questions:
If the goal is to win the car, is it in your advantage to stick with your original choice of door number 1, or to switch to door number 2, after the host has already shown you one goat. Or is the probability of winning the same regardless of if you switch or not? JUSTIFY YOUR REASONING WITH SENTENCES.
And what specifically is the probability (as a percent) of winning for each of those scenarios?
Explanation / Answer
It is better to switch because of the following reasons
If you stay with the door you picked initially you succeed if the initial door has a car, which has a chance of 1/3. If you’re strategy is to switch then you succeed if your initial pick is a goat, which has a chance of 2/3.
OR
The combinations of your first choice are:
Car
Goat(1)
Goat(2)
That’s a 1/3 chance of winning the Car – as you would expect.
After you make your first choice the combinations behind the remaining two doors are:
Goat(1) & Goat(2)
Car & Goat(2)
Car & Goat(1)
If Host Always picks a Goat then the remaining door contains:
Goat (1 or 2)
Car
Car
Hence if you switch there is a 2/3 chance of picking the Car.
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