The "Monty Hall Problem" applied to double-headers

[Mendoza Line]

On my mind because the second game of the Mil/Atl game is about to go.

This may belong in the Technicapping forum, but I never go in there, and this may get wider attention and discussion here.

For many, many years, I have used a handicapping tool I apply strictly to MLB double-headers and NHL regular-season no-rest home-and-homes, as it is fairly easy to consider those two games a closed set. (I believe MtrCityPimp does the same with MLB, just based on watching his plays.)

Anyone who has seen the old and new versions of the game show Let's Make a Deal might know that when the contestant chose one of three doors at the end of the original version, they would then show the contestant what was behind one of the doors they didn't choose, and then ask them if they wanted to change their pick. On the new version, they don't do that anymore. Why?

The controversial "problem" began when Marilyn vos Savant published a puzzle in her Parade Magazine column. One of her readers posed the following question:

“Suppose you’re on a game show, and you’re given a choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say number 1, and the host, who knows what’s behind the doors, opens another door, say number. 3, which has a goat. He says to you, ‘Do you want to pick door number 2?’ Is it to your advantage to switch your choice of doors?”

Ms. Savant, who’s listed in the Guinness Book of World Records Hall of Fame for “Highest IQ” (228), answered “Yes.” Because of the estimated 10,000 letters she received in response, she published a second article on the subject. A scholarly analysis of the letters she received was compiled by Donald Granberg, Ph.D., in a paper entitled "The Monty Hall Dilemma: To Switch or Not to Switch."

Due to the fervor created by Ms. Savant’s two columns, the New York Times published a large front page article in a 1991 Sunday issue which declared: “Her answer... has been debated in the halls of the C.I.A. and the barracks of fighter pilots in the Persian Gulf. It has been analyzed by mathematicians at M.I.T. and computer programmers at Los Alamos National Laboratory in New Mexico. It has been tested in classes ranging from second grade to graduate level at more than 1,000 schools across the country.”

The problem has been used as material for the CBS drama series NUMB3RS, the movie 21, and several books.

One of my personal favoites, The Drunkard’s Walk by Leonard Mlodinow provides readers with a guide to how the mathematical laws of randomness affect our lives. I think this should be required reading for serious sports bettors. He uses the Monty Hall Problem as an example of the "law of sample space."

Here is the formula illustrated:

Let the doors be called X, Y and Z.

Let Cx be the event that the car is behind door X and so on.

Let Hx be the event that the host opens door X and so on.

Supposing that you choose door X, the possibility that you win a car if you then switch your choice is given by the following formula:

P(Hz ^ Cy) + P(Hy ^ Cz) = P(Cy) P(Hz Cy) + P(Cz) P(Hy Cz) = (1/3 1) + (1/3 1) = 2/3



The simple way I explain the how to use this in MLB double-headers is this:

(As we all know, by the end of the MLB season, the final percent of over/unders will be approximately 50%. It's always close, much like the heads/tails result of flipping a coin a few thousand times.)

What are the potential outcomes for the totals in a 2 game set?

Over, Over
Over, Under
Under, Over
Under, Under

Let's now say that you know that the result of one of the two games is definitely "over," as in Game 1 of today's Mil/Atl double-header. What does that leave as the possible result combinations by the time both games are complete?

Over, Over
Over, Under
Under, Over

In only 1/3 of those sets are both results an "over" and in 2/3 of those sets the unknown game has to be an "under." The result of the game you did not know prior to the completion of both games is "under" two-thirds of the time.

I play these all the time, calling them "reverses" in my play threads, and this is what I'm talking about. The result of the 2nd game has a 66% chance of being the "reverse" of the result of the 1st game, all else being equal.



Comments welcome.

Thoughts, MtrCityPimp?

[Kacknballz]

[ZoomVT]

sorry man, not sold on it. I am familiar with this problem as it is similar to the old Bertrand's box problem. The answer is correct that you should switch doors, but this does not apply to wagering IMO.

In the case of sports there are external factors that affect the probabilities of each outcome, rendering the basic math principles useless. Just because there are 6 possible results, it does not mean that each has equal probability, therefore, just because you know 1 result, the probability of the other 3 outcomes does not change, is still tied to external factors.

If you are seeing success in this theory it is due to other reasons such as the winning team put more effort in first game, therefore wont play as hard second game. Or 1 team had their ace in one game and in the second game, they have a lower quality pitcher. Another possibility is one team used up their bullpen and is not available in the second game, etc.

[Mendoza Line]

Well, as I said, Zoom, it is "all else being equal."

Yes, there are external factors, but just as over-unders roughly find their way toward 50/50 over the course of a large sample (say, an entire season), this finds it's way toward 66% success over a large sample.

The easiest way to distort the formula when it comes to sports betting, would be if the oddsmakers put out wildly inaccurate totals over a long period of time. If you accept oddsmakers' totals to be reasonable, it will find it's way to 66% success.

[ZoomVT]

The missing part is how these events are correlated. how is this different than a night/day back to back? or how is it different than 2 random consecutive games.

Is there a mathematical or logical reason for this?

I am honestly curious not just being a dick.

[Mendoza Line]

Zoom --

I understand. And I think I indicated, maybe indirectly, that the two events are not correlated in any other way than it's an easy way to isolate two events as a set. In an isolated event with the possibility of two outcomes, you will be able to guess one of the outcomes with 66% accuracy if you know the result of the other outcome.

Any way that you can design a set of two events with a coin-flip's chance of success, you will have a long-term 66% chance of guessing the outcome of one event when knowing the outcome of the other.

Now if you accept this, and you should, because this is the data heading into today's play...

Non-Extra Inning Games: 188 overs (50.00%), 188 unders (50.00%)

...you know that MLB o/u's are coin-flip accurate.

One could choose to select the two games of a two-game series, games played by NL teams on two consecutive Sundays, games played agasints teams from Chicago on Thursdays....it doesn't matter.....the closer you get to an infinite number of these selections, the closer you get to 66% success. The math eventually will not care who the starters are, or what the wind conditions are, or what any other outside influence tries to say about the event results.

As I said before, the only thing that could negatively affect the results are inaccurate oddsmaker lines, and I'm talking really inaccurate, like putting a total of 2 on both halves of every double-header. Since we've all seen games with a 9.5-run total end with a 1-0 final, or games with 6.5-run totals end 13-10, you know that any result is possible. But, as you can see by the season-to-date results above, the oddsmakers put out pretty good lines over the long term, and in the end, they find their way to 50/50.

[Mendoza Line]

BTW, the 2nd game in Atlanta went over by 1 run with 2 outs in the bottom of the 8th. (My man Huddy with a 1-hitter, still blowin' it by 'em at 90+ in the 9th.)

Trust me, I wish I published this on a night it actually paid off.

But, as I tell people when I'm out and they ask me to give them a pick, "No guarantees. I don't win 'em all, just most of 'em."

In this case, only 66.6% will win, not 100%.

[MCP313]

I like to play a decent home team to avoid a sweep in a DH and usually I'll play opposite the total of game 1 in game 2 if the 1st game is a clear cut Over or Under.

I'll play the opposite result also in an identical SP matchup within 2 weeks of each other I.E Wolf/Wandy Rodrig. I faded Wolf I wanna say last week Friday or Saturday but I played the under both times....so there is nothing absolute about it.

So basically yah I kinda use it while administering my own filters....didn't play Braves Under game 2 tonight because I just had a hunch.

[Mendoza Line]

Originally Posted by MtrCtyPimp

I like to play a decent home team to avoid a sweep in a DH and usually I'll play opposite the total of game 1 in game 2 if the 1st game is a clear cut Over or Under.

I'll play the opposite result also in an identical SP matchup within 2 weeks of each other I.E Wolf/Wandy Rodrig. I faded Wolf I wanna say last week Friday or Saturday but I played the under both times....so there is nothing absolute about it.

So basically yah I kinda use it while administering my own filters....didn't play Braves Under game 2 tonight because I just had a hunch.

Yeah, I could tell you used it more selectively.

I get an opinon on game 2, so I know what I hope happens in game 1, and I'll maybe play a little more if it's on what I wanted anyway.

Your hunch was correct tonight, but by the skin of your teeth. An avoided loss will buy you roughly .003% advantage in the long run. But as you probably can figure, a missed win would cost you more.

I try to play certain plays (the ones I consider to be stand-alone plays outside of my normal handicapping) in a regimented way. I find that sticking with the formula will sometimes save me from myself. And I never worry when I lose, as I know what the eventual winning percentage will be when I'm done 50 years from now.

[tinwork]

Originally Posted by MtrCtyPimp

I'll play the opposite result also in an identical SP matchup within 2 weeks of each other I.E Wolf/Wandy Rodrig. I faded Wolf I wanna say last week Friday or Saturday but I played the under both times....so there is nothing absolute about it.

We've got this situation happening in Washington today, Sanchez v. Hernandez. They played last Sunday, with Florida getting the win 8-0. I understand that this isn't some sort of steel-tested system, but just wanted to give you a heads up in case you miss it. Anyway, I'm going to take Washington for a small piece just to see what happens.

GL!

[Mendoza Line]

Originally Posted by tinwork

We've got this situation happening in Washington today, Sanchez v. Hernandez. They played last Sunday, with Florida getting the win 8-0. I understand that this isn't some sort of steel-tested system, but just wanted to give you a heads up in case you miss it. Anyway, I'm going to take Washington for a small piece just to see what happens.

GL!

You'll also be taking Washington with Doug Eddings umping, and home teams are 8-1 with him this year. Nice to have something else to back up the reversal play on the Nats. GL!