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Ask the Slot Expert: Monty Hall on Survivor?

One of the twists in the new era of Survivor is the journey, in which one castaway from each tribe is taken to another island and given a task to do. The castaway usually leaves the island with an advantage or a disadvantage.

On a recent episode, the task involved choosing scrolls from a bag. One scroll is an advantage; the other two are lose-a-vote.

Each player must draw at least once. If they draw a lose-a-vote, they have the option to draw again. If they draw the advantage, the lose-a-vote is canceled and they get the advantage. If they draw the second lose-a-vote, they lose their vote at two tribal councils.

On their first draws, one player drew the advantage and the other two a lose-a-vote. They both decided not to tempt date and risk losing a second vote. They declined to draw again.

I wondered whether this game illustrated the Monty Hall problem and they should have drawn again.

First, what is the Monty Hall problem?

Do you remember on Let's Make a Deal when Monty Hall (or Wayne Brady for those not in AARP) would play the "three doors" game with a contestant? A really great prize is behind one of the doors and zonks, like a year's supply of cabbage, is behind the other two. The contestant picks a door, then Monty reveals what is behind one of the two doors not chosen. He then gives the contestant the option to switch to the other unchosen door.

Should the contestant switch?

The answer is yes. There is a one-third chance that the prize is behind the door originally chosen and a two-thirds chance that it is behind the other door.

Before you write to me to say that I'm crazy and that the chances are 50/50, you have to remember the original scenario.

When the contestants choose a door, they are dividing the sample set into two groups -- group 1 with one door and group 2 with two. (That worked out well.) We don't know where the prize is, so there's a one-third chance the prize is in group 1 and a two-thirds chance it is in group 2.

One thing we do know for sure is that at least one of the doors in group 2 is a zonk. Revealing what's behind one of the doors does not give us any new information, so there's no reason to change the chances of the prize being in group 2 from two-thirds to something else.

Now, if Monty were to pick another contestant and ask them to pick a door, that new contestant does have a 50/50 chance. But this scenario is not the same as the original scenario because the new contestant never had three doors from which to choose.

You can do an experiment to verify that the contestant should switch. Take three slips of paper. Write a P on one and a Z on the other two. Draw one at random. That's the door the contestant chose. Now look at the slips and note whether the P was on the one you chose (group 1) or on one of the other two (group 2). Repeat. The chances that the slip you chose has the P will approach 1/3. If you look at what is written on all of the slips, you can see that you can reveal a Z in group 2 without affecting the one-third chance that the P was in group 1.

If the probability of the prize being in group 1 is always 1/3, then the probability of the prize being in group 2 is always 2/3.

After a contestant drew a lose-a-vote (zonk) on Survivor, I wondered whether this is the Monty Hall problem all over again and they're better off drawing a second time?

I remembered my warning above about different scenarios and realized that the Survivor game wasn't the Monty Hall problem redux. The two scenarios are not the same.

The contestant's choice was revealed on Survivor. That is new information that affects the chances that the prize is still in the bag. The Survivor game is a completely different game.

After revealing that they had drawn a zonk, the contestant has a 50/50 chance of drawing another zonk or the advantage. I don't know how to put a numerical value on a lose-a-vote or an advantage, so I can't calculate the Expected Value of taking the L or drawing again.

The math of Sampling Without Replacement applies in the Survivor game.

Now, if Jeff Probst had magically appeared on the island after the contestant drew a scroll and said, "Wait. Before you open that scroll, let me open one of the scrolls you didn't choose," then the game would be the same as the Monty Hall game. But that didn't happen.

In this game, Monty Hall did not visit Survivor island.

If you would like to see more non-smoking areas on slot floors in Las Vegas, please sign my petition on change.org.

Send your slot and video poker questions to John Robison, Slot Expert™, at slotexpert@slotexpert.com.

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