Probability and Statistics The Monty Hall Problem Probability and Statistics Probability Theory

History September 1991, a reader of Parade asked a question to the “Ask Marilyn” column If you’re on a game show and you can choose one of three doors where there’s a car behind one and a goat behind the other two, after picking a door, would you switch doors after being revealed one with a goat? Is there an advantage? Marilyn Vos Savant responded saying it would be better to switch, and there was a lot of controversy with this response Matthew Carlton, Cecil Adams, and Keith Devlin later gave their reasoning that aligned with Marilyn’s

History cont’d This dilemma was named after Monty Hall Monty Hall was the host of Let’s Make a Deal in the 1960s and 1970s 3 doors are shown and the contestant picks one One door has a car, the other two have nothing A door that wasn’t picked is opened to reveal it’s empty The contestant has a choice to stick with their door or change to the other one

The big question…. Should you switch???

Explanation The choice isn’t luck but based on probability 1/3 chance of picking the car at the beginning Once a door is eliminated, the chance of winning a car between the last 2 doors is NOT 50-50 Need to look at 2 options: Always switching Always staying

Explanation Cont’d Always stay: Always switch: 2/3 chance of picking a door with nothing 1/3 chance of picking the door with the car Always switch: 2/3 chance of picking a door with the car 1/3 chance of picking a door with nothing

There is a 2/3 chance of getting the car if you switch. This means you have a better chance at winning if you switch!

References http://math.ucsd.edu/~crypto/Monty/montybg.html https://www.khanacademy.org/math/trigonometry/prob_c omb/dependent_events_precalc/v/monty-hall-problem http://en.wikipedia.org/wiki/Monty_Hall_problem#Solution s