The True Count Theorem
by Abdul Jalib
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Exerpt take from the following reference in DejaNews:
Subject: Proof that everyone has the same shot at getting tens
From: [.. (Abdul Jalib M'hall)
Date: 1996/07/30
Message-ID: <..>
Newsgroups: rec.gambling.blackjack
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The True Count Theorem
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The following is a mathematical theorem:
Theorem: the expected value of the true count after a card is revealed and removed from any deck composition is exactly the same as before the card was removed, for any balanced count, provided you do not run out of cards.
See the proof at the end of the article. Expected value is a precise mathematical term defined as the mean average, which is computed by summing the probability of an event times the value of that event, over all possible events. So the expected value of the true count after drawing a card is the summation of the probability of drawing each card times the value of the true count after drawing that card.
One consequence of this theorem is that the expected value of the count after any number of cards have been revealed and removed will be the same as before, and so the expected value of the count after a round has been dealt will be the same as before. And when you get a constant number of rounds, the expected true count after each is zero. Since the true count starts at zero, the overall expected value of the count is zero, when you get a fixed number of rounds.
The Running Count
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The theorem applies only to the true count, not to the running count. The running count does not obey the same laws of as the true count.
With regards to the effects of other players at the table and the tendency of the round to stop with a big card, much confusion stems from a mistaken assumption that the behavior of the true count is the same as the behavior of the running count.
The running count must be zero at the end of the deck. Therefore, drawing cards in high counts tends to cause the running count to fall, and drawing cards in low counts tends to cause the running count to fall. But the expected true count is unchanged.
The Proof
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Theorem: the expected value of the true count after a card is revealed and removed from any deck composition is exactly the same as before the card was removed, for any balanced count.
Proof:
Let Wi be the Weight of the card of rank i, i.e., the count value.
Let Ni be the Number of cards of rank i already revealed, counted, and removed from the deck.
Let Si be the Starting number of cards of rank i in a full deck.
Let Li be the number of cards of rank i currently Left in the deck, i.e., Li=Si-Ni
Let C be the number of Cards remaining in the deck, i.e., C=sum_over_i{L}
Let R be the Running count.
Let T be the True count, i.e., T = R/C
Assume that the count is balanced, i.e., sum_over_i{W*S}=0.
Need to show that: T = sum_over_j{((R+W[j])/(C-1))*(L[j]/C)}, that is to say that the true count after a card j is revealed, removed, and counted, averages out to the same as the true count before. The average is computed by adding up the true counts after each card is drawn weighted by the probability of each card being drawn.
sum_over_j{((R+W[j])/(C-1))*(L[j]/C)} =
(1/(C(C-1)))sum_over_j{(R+W[j])L[j]} =
(1/(C(C-1)))sum_over_j{(RL[j]+W[j]L[j])} =
(1/(C(C-1)))sum_over_j{(RL[j]+W[j](S[j]-N[j])} =
(1/(C(C-1)))sum_over_j{(RL[j]+W[j]S[j]-W[j]N[j])} =
(1/(C(C-1)))(sum_over_j{RL[j]} + sum_over_j{W[j]S[j]} - sum_over_j{W[j]N[j]}) =
(1/(C(C-1)))(R*sum_over_j{L[j]} + 0 - R) =
(1/(C(C-1)))(R*C-R) =
(1/(C(C-1)))(R(C-1)) =
R/C = T
. . . QED
Again...
Theorem:
The expected true count after a card is revealed and removed from any deck composition is the same as before the card was removed, for any balanced count, provided you do not run out of cards.
Corollary:
The expected true count after any number of cards are revealed and removed from any deck composition is the same as before the cards were removed, for any balanced count, provided you do not run out of cards.
Corollary:
The expected true count after a round is the same as before the round, for any balanced count, provided you do not run out of cards.
