Neat question! Google search for the exact answer (3724430600965475/123979633237026, as Tom said) turns up a hit in Russian; translation at https://translate.google.com/translate?hl=en&sl=ru&u=https://habr.com/users/... Ah, and Google search for just the numerator yields a hit on https://www.jstor.org/stable/2687606. Registering for a free JSTOR account, that URL shows me that this was Problem #630, "Guessing Card Colors", appearing with solution in The College Mathematics Journal Vol. 30, No. 3 (May, 1999), pp. 234-235. The problem was posed my Michael Andreoli, Miami-Dade Community College (North), Miami, FL, and the published solution was by John Henry Steelman, Indiana University of Pennsylvania, Indiana, PA. Andreoli's formulation of the problem there presupposes that you always guess the more-likely color (and guess randomly when tied), which as Cris says is optimal. Steelman proves that for m red cards and n black cards with m >= n, the expected value is E(m,n) = m + sum{k=0..n-1} binomial(m+n,k)/binomial(m+n,n) The proof is by induction on m+n. When m=n, this simplifies to E(n,n) = n - 1/2 + 2^(2n-1)/binomial(2n,n) Ah: And there's an editor's note that one of the solvers (Michael Vowe of Therwil, Switzerland) pointed out that this is a special case of Problem #4661 from the American Math Monthly, vol 63 (1956), pp. 51-53. But I haven't tried to track down the AMM article to see what the generalization is -- maybe to drawing marbles out of an urn with more than 2 colors? --Michael On Wed, Dec 26, 2018 at 6:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
Although I believe this:
----- Hmm, strategy seems clear: guess the majority color remaining, if any -----
, I'm a lot less sure of the rarish case when it's a tie: Is it wiser to, say, alternate your guesses when it's a tie (to pick up, you know, a second-order advantage, now that you have maxed out your first-order advantage), or is it entirely irrelevant (to your expected gain) what you pick at those times?
Actually, now that I'm writing it, I kind of sort of *think* that it is completely obvious that it's entirely irrelevant.
On the other hand, suppose that instead of only considering the *expected* value of this game we view it as a 2-player thang.
You are playing against an opponent — each playing in isolation from the other — but with the winner being *not* necessarily the person who is ahead by the end.
Instead, what if the winner is defined to be *the one who was *leading* more*.
Where a scorekeeper has a running tally of each player's progress through the game. Each player gets one metapoint for each completed t'th stage during the ordinary game, 1 <= t <= 52, at which they are strictly ahead of the other.
If at the end of the game *leading more* (i.e., having the most metapoints) were the only thing that mattered, would that affect strategy at ties?
Now: Onward, to the third-order effects.
—Dan
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