Michael Malecki recently shared a link to a Business Insider article that discussed the Monty Hall Problem.

The problem starts with three doors, one of which has a car and two of which have a goat. You choose one door at random and then the host reveals one door (not the one you chose) that holds a goat. You can then choose to stick with your door or choose the third, remaining door.

Probability theory states that people who switch win the car two-thirds of the time and those who don’t switch only win one-third of time.

But people often still do not believe they should switch based on the probability argument alone. So let’s run some simulations.

This function randomly assigns goats and cars behind three doors, chooses a door at random, reveals a goat door, then either switches doors or does not.

monty <- function(switch=TRUE)
{
    # randomly assign goats and cars
    doors <- sample(x=c("Car", "Goat", "Goat"), size=3, replace=FALSE)

    # randomly choose a door
    doorChoice <- sample(1:3, size=1)

    # get goat doors
    goatDoors <- which(doors == "Goat")
    # show a door with a goat
    goatDoor <- goatDoors[which(goatDoors != doorChoice)][1]

    if(switch)
        # if we are switching choose the other remaining door
    {
        return(doors[-c(doorChoice, goatDoor)])
    }else
        # otherwise keep the current door
    {
        return(doors[doorChoice])
    }
}

Now we simulate switching 10,000 times and not switching 10,0000 times

withSwitching <- replicate(n = 10000, expr = monty(switch = TRUE), simplify = TRUE)
withoutSwitching <- replicate(n = 10000, expr = monty(switch = FALSE), simplify = TRUE)

head(withSwitching)
## [1] "Goat" "Car"  "Car"  "Goat" "Car"  "Goat"
head(withoutSwitching)
## [1] "Goat" "Car"  "Car"  "Car"  "Car"  "Car"

mean(withSwitching == "Car")
## [1] 0.6678
mean(withoutSwitching == "Car")
## [1] 0.3408

Plotting the results really shows the difference.

require(ggplot2)
## Loading required package: ggplot2
require(scales)
## Loading required package: scales
qplot(withSwitching, geom = "bar", fill = withSwitching) + scale_fill_manual("Prize", 
    values = c(Car = muted("blue"), Goat = "orange")) + xlab("Switch") + ggtitle("Monty Hall with Switching")

qplot(withoutSwitching, geom = "bar", fill = withoutSwitching) + scale_fill_manual("Prize", 
    values = c(Car = muted("blue"), Goat = "orange")) + xlab("Don't Switch") + 
    ggtitle("Monty Hall without Switching")

(How are these colors? I’m trying out some new combinations.)

This clearly shows that switching is the best strategy.

The New York Times has a nice simulator that lets you play with actual doors.

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Jared Lander is the Chief Data Scientist of Lander Analytics a New York data science firm, Adjunct Professor at Columbia University, Organizer of the New York Open Statistical Programming meetup and the New York R Conference and author of R for Everyone.

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