Ask the Slot Expert: Monty Hall plays Survivor

The current and prior seasons of Survivor added a twist that illustrates the Monty Hall Problem.

Survivor has two types of challenges. In an endurance or last-man-standing challenge, the last person still standing on a buoy or balancing a ball on a plank wins. In a skill or finish-first challenge, the first person to complete a task or solve a puzzle wins.

The new twist in the past two seasons is forcing the survivor who finishes last in a skill challenge to play a game called Do or Die at the next tribal council. The survivor is shown three boxes. Inside one box is the image of a flame, which means the survivor is safe that night and cannot be voted out. The other two boxes contain an image of a skull, which means that the survivor is eliminated from the game.

This is the classic Let's Make a Deal situation. You have a choice of three boxes or doors. Monty (or, more recently, Wayne Brady) tells you that behind one door is a car or some other fabulous prize and behind the other two are zonks, like goats.

You make your choice and then the hosts reveals the zonk or skull inside one of the two items you didn't choose. The host then gives you the option to stick with your original choice or switch to the remaining item.

Should you switch? (Well, what do you think? Should you?)

Both times the game has been played on Survivor, the survivors stayed with their original choice and both times the boxes contained the flame and the survivor was safe.

Mathematically speaking, the players made a poor choice. They should have switched.

There are three choices. One is good and two are bad. The probability of choosing the good outcome is 1/3 and the probability of choosing the bad is 2/3.

You make your choice. There is a 1/3 probability that you have the good box. Each of the other two boxes also has a 1/3 probability of being good, so there's a 2/3 chance the good box is one of the boxes you did not choose. We also know that at least of of the boxes you didn't choose is bad.

The boxes are partitioned into two groups: one containing one box with a 1/3 chance of being good, and another containing two boxes with a combined 2/3 chance of being good.

Jeff (or Monty or Wayne) reveals one of the boxes you didn't choose. It is bad. That is not new information. We already knew that at least one of the boxes you didn't choose was bad.

Some people may think that the probability that their box is good has increased to 50%, but it's still 1/3 because we were not given any new information. What has happened, though, is the probability that the remaining box you did not pick is now 2/3.

You have a 1/3 chance of having chosen the good box and the other box has a 2/3 chance of being good. That's why you should switch.

Let's call the box you chose Box 1 and the other two Box 2 and Box 3. The following table shows what could be in each box and what happens if you stay with your original pick or switch.

Box 1	Box 2	Box 3	Stay	Switch
G	B	B	Win	Lose
B	G	B	Lose	Win
B	B	G	Lose	Win

You win 2/3 of the time when you switch and only 1/3 of the time when you stay. If only we could know when it's that 1/3 time when we should stick with our original choice. Because we don't know, we play the probabilities and switch.

Here's another way to look at it. If you're going to stay with your original choice, the only way you win is if you chose the good box and you had a 1/3 chance of doing so. The other 2/3 is with the other two boxes and the host eliminated one of the boxes for you.

The outcomes of the games played on Survivor illustrate a consequence of statistical decision making. Sometimes bad decisions lead to good outcomes. Conversely, sometimes good decisions lead to bad outcomes.

The game also illustrates an aspect of probability that many people don't understand. Here's a transcript of a segment from the November 6, 2020 episode of Tooning Out the News. In the segment they played a clip from ABC's election night coverage in which George Stephanopoulos talks to pollster Nate Silver.

GS: Let's bring in Nate Silver from FiveThirtyEight. Nate, talk about the forecast you had coming into election night.

NS: So, coming in we had Joe Biden with an 89% chance of winning, Trump with 10%, 1% chance of a tie. [edit] So in 2016 we had Trump with a 30% chance of winning, roughly.... [edit] But look, you know, I don't control when the 30% happens, the 10% happens, so there--

At this point, an animated reporter walks into frame and puts an animated sash that says Mr Wrong 2020 on Mr. Silver.

Reporter: Congratulations to Nate Silver, the new Mr. Wrong 2020. As Mr. Wrong, you will write a think piece about why you were actually right before the news media gives you another chance in 2024.

Thirty percent, 10 percent, 1 percent, whatever, are not 0 percent. You have about a 0.0025 percent chance of hitting a royal flush, yet we all hope to hit one and 99.999 percent of us have hit one or more.

Gamblers understand that even a small chance is better than no chance and they never know when an event will occur. Probability may say that you're better off switching, but sometimes you win when you don't. Nate Silver was not wrong.

Click here for the latest Covid data.