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## Reader Comments and Retorts

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1. andrewberg posted on March 26, 2014 at 02:22 PM #Statements posted here are those of our readers and do not represent the BaseballThinkFactory. Names are provided by the poster and are not verified. We ask that posters follow our submission policy. Please report any inappropriate comments.That's the bet example he could come up with for things that seemed obvious?

I absolutely disagree with that assertion...It is very counter intuitive. If he is saying young means 26-30...then sure I'll agree there, but until someone produces a study on consistency, it is more obvious that young players are inconsistent. (again young is under 26 years old)

It seems obvious to me that you should switch, what am I missing?....

I think you're right for hitters, but I thought there were studies that showed pitchers are pretty much what they are from the get go.

Your own column in Parade magazine.

Here is the game show problem explained by von Sant (and debated by readers).

The host then eliminates *one of the wrong doors*. If you originally chose incorrectly, which you would have done 2/3 of the time, then you always win by switching. If you originally chose the correct door, which you would have done only 1/3 of the time, you always lose by switching.

Therefore, if you don't switch, you win 1/3 of the time. If you do switch, you win 2/3 of the time.

It's not obvious to me at all, what I think is that very inconsistent young players tend to become more consistent over time- very consistent young players become less consistent over time- why? 1 part regression to the mean, 1 part simple change-

The young consistent player will get banged up as he gets older, he'll heal slower so that physically he will be different from one week to another, one month/year to another

the inconsistent player likely does not have uniform approach, he'll fix things whether they need to be fixed or not, his effort or lack thereof will be inconsistent, his attention span will wander, that type of thing will tamp down over time.

Of course I don't know either-

the thing is James used to throw these things out, he'd reason himself into something- but then he'd TEST it, he'd devise a study, and if he as wrong he'd say so in the next year's Abstract.

A great one was a blurb he wrote on Lloyd Moseby after Moseby's seeming break out year in 1983- he said that looking at, when a player has a clearly established skill level, not an erratic performer, but one with a clearly established level, and then one year plays at a completely different and higher level- isn't that evidence that the change is real? Seems obvious to me, isn't it obvious to you? It wasn't obvious to me, to me Lloyd Moseby going from hitting .235/.290/.360 every year to .315/.376/.499 was obviously a fluke... 20/20 hindsight, 1983 was to be Lloyd's best year, but 1984-91 he did play much better than he had pre-1983.

I don't think I've ever seen anything like that. Most studies I have seen have shown that pitchers prime seasons is older than hitters, and everything I've seen from personal observation has indicated that young pitchers are massively inconsistent. There just doesn't seem to be many pitchers like Maddux/Clemens who can maintain any high consistency for any length of time, most are like less extreme versions of Saberhagen. Even within the season, I think you'll see more inconsistent performances from younger than you will from established veterans.

When I was young and for about the one day I was into bodybuilding, there was talk about why there were few if any championship bodybuilders under 25 years old, and the argument was that young bodies are very elastic from a day to day basis, and it's hard to maintain consistent mass/tone because of youth. Extending that out to athletics, and it can mean your muscles change more rapidly on a day to day basis affecting the consistency. This always made sense to me and I don't think I have ever seen anything contradicting this.

A simple illustration is to consider the problem with 1000 doors. You pick a door, Monte Hall opens 998 doors and asks if you want to switch. 999 times out of 1000, your initial guess was wrong and you should switch. 1 time out of 1000, your initial guess was right, and you lose out on the prize by switching. The 3 door problem is exactly the same, except the probabilities are 2/3 and 1/3.

Most people get it wrong because they think it's a 50/50 proposition with the second choice, because one of them wins and one of them loses -- they forget that 2/3 of the time they were wrong with their first choice, and Monty opening a losing door doesn't change that -- because he was always going to open a losing door.

Maybe I was thinking of this which says pitchers peak in strikeout rates at 24, but that doesn't necessarily speak to overall performance and consistency.

It's a good study though. Lots of information there.

Why does it matter? This doesn't change the odds of the second choice at all. It's still a 50/50 proposition.

In fact, because Monty was always going to show an empty door, and because the contestant always gets the second chance, it's wrong to think about the first choice as being a 1/3 proposition. Really the first choice is wholly irrelevant. There's no logic as to why a contestant should automatically change their guess - it's going to be 50/50 either way. When it comes down to it, this isn't really a game about picking among three doors, it's a game about picking among two doors.

Yep, that's exactly the incorrect logic that fools people.

Try it yourself. Use a deck of cards, 52 doors, try to find the Ace of Spades. Then have somebody else (who can look at them) "open" 50 of them that aren't the Ace of Spades and see if you think you should switch then...

Why does it matter? This doesn't change the odds of the second choice at all. It's still a 50/50 proposition.No, it's a 2/3 : 1/3 proposition.

If the initial guess was correct, switching loses the prize. This happens 1/3 of the time.

If the initial guess was incorrect, the chosen door is empty. Monte Hall shows you the other empty door, which means that switching wins the game. This happens 2/3 of the time.

Conclusion: it's better to switch.

The broken symmetry comes from the fact that Monte only shows you an empty door

which you did not initially choose.Thus, he is giving you information about the door you didn't choose, but not about the door which you did choose (since he will never open your door to show you whether it is empty or not).If Monte could open any empty door, including the one you chose, your 50:50 intuition would be right. The trick to the problem is that he is pretending to do that, but is actually doing something else, and that's why so many people get it wrong.

thank you, I was trying to find his quote on it.

If you switch and you're wrong, look on the bright side- the taxes on winning a goat are a lot less than those of winning a car.

Do not confuse the puzzle with the actual mechanics of the show Let's Make a Deal. And also do not confuse Monty Hall with a mathematician.

No. It is 2/3 to switch only if the host is only allowed to open a losing door.

Scenario 1: Host always opens a losing door, if you switch you win 2/3.

Scenario 2: Host randomly chooses a door to open. 1/3 of the time you lose before you get the option to switch. If you switch, you win 1/3 of the time (or 1/2 the time conditional on a losing door being opened). This scenario operates exactly how most people think the problem does. Scenario 2 is Deal or No Deal, with the contestant randomly choosing, by the way.

Scenario 3: Host only opens a losing door when you've chosen correctly, hoping to sucker you into switching. If you switch, you lose 100% of the time.

The original question:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?There is absolutely nothing in the question that says the host is required to open a losing door (scenario 1). Vos Savant's answer, with added emphasis:

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors andwill always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?So she poses the correct scenario where switching makes sense but there's nothing in the original scenario that justifies her assumption. The situation you are in is that you have chosen one door and the host has opened an empty door. Short of other external information, this hasn't changed one damn thing. You have to know what rules the host is operating under.

In short:

Scenario 1: 2/3 win if switch

Scenario 2: 1/3 win if switch

Scenario 3: 0/3 win if switch

The likelihood of each scenario is unknown. (It could of course be "known" if in the past 1000 shows, the host has opened a losing door and offered a switch.) If the scenarios are equally likely then the odds are 1/3 if you stick and 1/3 if you switch. The only way it makes sense to switch is if Scenario 1 is more likely than Scenario 3. If 3 is more likely than 1, you should stick. Given a game show host is trying to maintain/increase ratings and assuming audiences like to see contestants win fairly frequently, it might make sense to assume that 1 is more likely than 3. However, if it's some game run by a casino or the state lottery, you can assume they want you to lose more than win (therefore depends on the payout odds). If if it's a street 3-card Monte hustler, 3 seems more likely than 1 (of course the real scenario there is that he palmed the red queen).

By game theory, the host should mix things up a bit to keep contestants guessing but, if his goal is for you to lose more often than win (while maintaining interst), his optimal move is certainly not 100% Scenario 1.

So in essence, the reflexive 50/50 crowd was wrong and vos Savant cheated. If the only information you have is that the host opened an empty door and you have NO information about the host's behavior, then it doesn't matter whether you switch or not.

The various knowledges involved here remind me of the similar Two Children problem (I have 2 kids, one's a boy, what are the odds the other's a boy?) It depends on what one knows about how my family got selected for this puzzle in the first place.

It feels like the host is giving you no information, because you knew that one of the unchosen doors, at least, was a goat. That's what makes the problem so wonderful. You can argue all you want, but it works to switch. Explain it to yourself however you want. It works.

isthe better of the other two doors. If you think about it precisely that way - that he is offering you the better of doors 2 and 3 - then it's obvious you should switch.To drive this home, let's say the problem went more strictly as I put it. You choose Door #1. The host, without revealing anything, then offers you the chance to trade Door #1 to get Door #2 plus Door #3 minus a goat. (That is equivalent to "the better of the two doors".) Do you switch? Hell yes. There's a 2/3 chance that works out to a car, and a 1/3 chance it works out to a goat.

Now let's say before the switch offer the host says, "What if I told you at least one of those two doors has a goat? Do you want to keep Door #1?" Hell no. There's a 100% chance at least one of those two doors has a goat; this was true when you chose Door #1, and it's true if you switch to Door #2 + Door #3 - goat. The host stating that one has a goat tells you nothing. The host testifying under oath that one has a goat tells you nothing. The host

opening up one of the doors to reveal a goattells you nothing.In the problem as originally stated, that's basically what is going on. You choose Door 1. The host offers you the chance to switch to the better of the other two doors, which he has narrowed down by showing the worse of the two. There's a 2/3 probability the better of Doors 2 and 3 is a car. Switch.

neverthe car. So, yes, without prior knowledge we don't know that the door being revealed won't be the car. But this is kind of like saying without prior knowledge we don't know that an unopened Coke bottle contains Coke. While that's true on a literal level, we can also say thatwithprior knowledge we can safely assume Scenario 2 is unlikely. (The same can be said for Scenario 3.)What if you're trying to win a goat?Indeed.

You're stealing Bill James' bit.

You can choose door #1, or

You can choose to open doors 2 and 3 and keep whatever is behind either.

That's essentially what happens and should make it a bit more obvious.

Yes, Vos Savant presented the problem with incorrect rules and that's where all the confusion comes from. She assumes that Monty always opens one unchosen losing door. In that case the contestant should switch because his original door is 1/3 and the remaining closed door is 2/3.

But that doesn't hold true if Monty can immediately open a chosen losing door to go "ha ha you lost" without offering the chance to switch. So he opens another door only if the original choice wins. In that case the contestant should obviously not switch.

And by all accounts, the real show held Monty to neither of those sets of rules, but let him do whatever he wanted. Then there is no mathematical solution at all but you have to outguess Monty. Like a poker bluff.

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