DoND expected value: average or median?

[Blaq]

The Wikipedia entry for Deal or No Deal states:

The expected winning at any point in the game is the median of the unopened boxes.

What gives? It's the average, right? (Blaq suddenly feels like everything he knew is wrong...) Even the dictionary defines "expected value" as

The sum of all possible values for a random variable, each value multiplied by its probability of occurrence.

Since all remaining amounts bear the same probability, this sum of products is simply the average.

If nothing else, I can disprove the "median" claim with this absurd proof:

Say the unopened cases are $10, $100, and $1,000. According to Wikipedia, the expected value would be $100.

Now say the unopened cases are $10, $100, and $1,000,000. It's obvious the expected value is much higher. According to Wikipedia, the expected value would still be $100. Put a Google dollars in the last case, the game is worth the same. Rubbish.

[clemon79]

[quote name=\'Blaq\' date=\'Mar 8 2006, 11:57 AM\']What gives? It's the average, right? (Blaq suddenly feels like everything he knew is wrong...) Even the dictionary defines "expected value" as
[/quote]
Yes. Whoever wrote that article apparently wasn't paying attention in algebra class. You have it just right. The median is the "middle" value (or the average of the two middle ones when there is an even number of members) when ordered, whereas the mean (the "average") is the sum of the members divided by the number of members.

[Gus]

Since neither of you decided to be bold, I went and corrected the article.

[clemon79]

[quote name=\'Gus\' date=\'Mar 8 2006, 03:12 PM\']Since neither of you decided to be bold, I went and <gasp> corrected the article.

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Blaq wasn't sure he was exactly right and wanted to be sure he was, which is a really good tack to take when deciding whether to correct a Wikipedia article, and one the original author would have done well to take as well.

And I was running around like a chicken with my head cut off this morning getting ready for a job interview, so I didn't have time to go do it.

So, yeah, save the <gasp> for a time when it's warranted.

[davidhammett]

[quote name=\'Blaq\' date=\'Mar 8 2006, 02:57 PM\']The Wikipedia entry for Deal or No Deal states:

The expected winning at any point in the game is the median of the unopened boxes.

What gives? It's the average, right? (Blaq suddenly feels like everything he knew is wrong...) Even the dictionary defines "expected value" as

The sum of all possible values for a random variable, each value multiplied by its probability of occurrence.

Since all remaining amounts bear the same probability, this sum of products is simply the average.

If nothing else, I can disprove the "median" claim with this absurd proof:

Say the unopened cases are $10, $100, and $1,000. According to Wikipedia, the expected value would be $100.

Now say the unopened cases are $10, $100, and $1,000,000. It's obvious the expected value is much higher. According to Wikipedia, the expected value would still be $100. Put a Google dollars in the last case, the game is worth the same. Rubbish.

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Well, of course I agree with the "mean" being the relevant calculation if expected value is the concern.  The problem is that expected value is only one way of analyzing the situation.  Expected value is essentially an indication of what will happen "on average"... which means that, over time, each game with those circumstances will give away that amount of money per game.  Of course, some of those games would give away more, some less.  (And of course, the chances of having more than one game with exactly the same set of remaining values are pretty slim in the first place.)

But for a contestant who is playing the game one time, expected value loses some of its relevance.  It does provide some guidance; i.e., if the deal is less than the expected value, then the possibility of a higher amount makes it worth not taking the deal.  Of course, this is where individual situations and such take over on the part of the contestant:  Do they need the money?  Are they a gambler?

However, the fact that the expected value of having a $10, $100, and $1000 case left over works out to be $370 (so a smaller deal should persuade you to play on) may not mean as much as the fact that the chances are that the one case you have is one of the smaller values.  Thus, the median provides some sort of guidance as to what could happen -- not on average, perhaps, but in the individual contestant's case.  And of course, Blaq's point is well taken... the deal that would be offered for $10, $100, and $1M would likely be in the hundred thousands, significantly higher than the median.

My apologies for the rambling, but it's one of the elements of the show that still keeps me thinking... about the mathematics, about the psychology, and about the phenomenon that has (at least temporarily) taken hold of a decent chunk of the country.