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What mistake are these lottery players making in their logic here?
They are thinking that there are a finite number of numerical combinations,
and that when one is drawn it is discarded and therefore removed from
the game which continues through to the next draw (the mistake lies
in this latter supposition). They are thinking that the numbers have "memory".
Of course they are wrong, the odds for any one number out of fifty to
be drawn randomly is, surprisingly, one in fifty; the odds for any particular
six numbers to be drawn randomly from fifty are about one in fourteen
million, and from draw to draw that never changes. Ironically, though,
there is a game where when a number is drawn it is removed from the
game, so to speak, and the cards do have "memory". The faulty
logic of the lottery players is correct here. This is, of course, the
game of blackjack!
Imagine that a standard deck of cards is shuffled and a hand of blackjack
is played. The cards that were played in the hand are then taken out
of play and not returned to the deck of cards which is then used to
deal the next hand to the players. The odds in favour of the player
have changed – possibly for better, possibly for worse – depending upon
which cards have been removed from the game. For example, if the discarded
cards were mainly low-value cards, then the odds move in favour of the
player, because that means they player is a little more likely to receive
a high-value card during the next hand and so he will probably get closer
to blackjack. A player can, and should, take note of what is happening
with the cards and bet accordingly. What makes the game even more delightful
is that even after the player has bet, there is usually still a further
test of skill for him to face – the decision as to whether to "twist" (ask
for another card) or stick with his current hand in the hope it will
be enough to beat the other players. He has to estimate not only the
likelihood that one of the other players is holding a higher value hand
than his, but also whether the remaining cards are tending more towards
higher or lower values. Then he makes the final call, and the hand ends
elegantly with the inevitable moment of truth.
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