Author Topic: A Twist on the Monty Hall Problem (Read 4147 times)

chris319 · « **Reply #15 on:** March 30, 2011, 02:58:50 AM »

My conclusion makes all kinds of sense if you are familiar with the Law of Large Numbers, viz.:

Quote

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

The LLN is important because it "guarantees" stable long-term results for random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins.

http://en.wikipedia.org/wiki/Law_of_large_numbers

In this case the expected value is 66.6% wins for switching. One independent trial, as on a TV game show, is an insufficient number of trials for the Law of Large Numbers to give the player an edge. If the player had a larger number of opportunities to play, the odds would approach 66.6% for switching.

Below are the results of 10 Monte Carlo simulations. The results are anomalous until we get up to five trials. One million trials give the expected 2:1 odds:

Trials: 1

No switching
Wins: 0
Zonk: 1
0% wins

Switching
Wins: 1
Zonk: 0
100% wins

Trials: 2

No switching
Wins: 1
Zonk: 1
50% wins

Switching
Wins: 1
Zonk: 1
50% wins

Trials: 3

No switching
Wins: 0
Zonk: 3
0% wins

Switching
Wins: 3
Zonk: 0
100% wins

Trials: 4

No switching
Wins: 2
Zonk: 2
50% wins

Switching
Wins: 2
Zonk: 2
50% wins

Trials: 5

No switching
Wins: 1
Zonk: 4
20% wins

Switching
Wins: 4
Zonk: 1
80% wins

Trials: 6

No switching
Wins: 2
Zonk: 4
33% wins

Switching
Wins: 4
Zonk: 2
67% wins

Trials: 7

No switching
Wins: 1
Zonk: 6
14% wins

Switching
Wins: 6
Zonk: 1
86% wins

Trials: 8

No switching
Wins: 2
Zonk: 6
25% wins

Switching
Wins: 6
Zonk: 2
75% wins

Trials: 9

No switching
Wins: 2
Zonk: 7
22% wins

Switching
Wins: 7
Zonk: 2
78% wins

Trials: 9

No switching
Wins: 2
Zonk: 7
22% wins

Switching
Wins: 7
Zonk: 2
78% wins

Trials: 10

No switching
Wins: 2
Zonk: 8
20% wins

Switching
Wins: 8
Zonk: 2
80% wins

Trials: 1000000

No switching
Wins: 334281
Zonk: 665719
33% wins

Switching
Wins: 665719
Zonk: 334281
67% wins

dale_grass · « **Reply #16 on:** March 30, 2011, 08:45:32 AM »

Quote from: chris319 on March 30, 2011, 02:58:50 AM

My conclusion makes all kinds of sense if you are familiar with the Law of Large Numbers ...

I'm familiar with the law of large numbers. I'm confused as to your using the law of large numbers and a simulation of thousands of trials to defend your position of "it's a one time shot." Maybe I'm mistaken.

Better yet, let me just ask this: If you were on the show looking at the guaranteed pair and Wayne asked if you wanted to switch, would you?

Neumms · « **Reply #17 on:** March 30, 2011, 09:12:24 AM »

Quote from: clemon79 on March 29, 2011, 06:11:27 PM

Quote
2. If they could have switched to one of the other three cards, would that probability change, and if so, to what?
Well, 1 in 4, ostensibly. Four cards left, one is right, three are wrong, they get to pick which one they want. As you said, they are shown their guaranteed pair, so everything up to that point is chrome.

That said, one chance in four to win a car is no better than your chances in many TPIR games for a car. The little games for a car that offered little chance to win (guess the suggested retail price of a can of paint on the West Coast within 50 cents, for example) got old even on Monty's version.

davidhammett · « **Reply #18 on:** March 30, 2011, 12:19:33 PM »

Quote from: dale_grass on March 30, 2011, 08:45:32 AM

Quote from: chris319 on March 30, 2011, 02:58:50 AM
My conclusion makes all kinds of sense if you are familiar with the Law of Large Numbers ...

I'm familiar with the law of large numbers. I'm confused as to your using the law of large numbers and a simulation of thousands of trials to defend your position of "it's a one time shot." Maybe I'm mistaken.

Same here. Certainly things will average out over time, and will not (in fact, cannot) exactly mirror the theoretical probabilities for the first case or cases. However, that does not change what the probability is... it only changes how you choose to interpret and use it.

Dbacksfan12 · « **Reply #19 on:** March 30, 2011, 03:16:03 PM »

Quote from: Neumms on March 30, 2011, 09:12:24 AM

The little games for a car that offered little chance to win (guess the suggested retail price of a can of paint on the West Coast within 50 cents, for example) got old even on Monty's version.

Yes, but in those situations, people could make a somewhat educated guess.

Mr. Armadillo · « **Reply #20 on:** March 31, 2011, 09:17:59 AM »

Even if it's still a one-time shot, you're going to want to do everything you can to increase your chances to win.

If I told you tomorrow that you'd have one shot to nail a half-court shot at halftime of the Butler-VCU game for a million dollars, you'd hit up a local basketball court and take a few practice shots beforehand, right? Same principle here. Anything you can do to get a few extra percentage points is usually worth it.