Mathematical_Recreations-Kraitchik-2e

Probabilities 129
deck is usually not used. Thus the banker can sell 29 cards
and must pay 27 times the price of each card, which gives
him an advantage of 29:27, or about 1.074.
10. BACCARA. The banker deals two cards to the punter
and two to himself. The punter may either stand pat or ask
for an additional card. The banker has the same rights, but
he has the advantage of knowing before he decides whether
the punter has asked for another card and what the card was,
in case the punter has taken a third card. Each face card
counts ten, and the others count according to the number of
their pips; if the sum of the cards in a hand equals or exceeds
10 or 20, the 10 or 20 is not counted; that is, in valuing a
hand, 10 or 20 counts for O. The winner is the one with the
highest total. If one of the players receives 8 or 9 with two
cards, no cards are drawn and the winner is declared on the
basis of the cards dealt.
J. Bertrand discussed the following question: If the punter's
first two cards give him a count of 5, should he ask for a
third card? Bertrand does not give a single direct answer.
His conclusions are:
(1) If the punter stands pat and the banker knows that he
usually does so, the respective probabilities that the player
will win, draw, or lose are 0.445, 0.086, 0.469.
(2) If the punter asks for a card and the banker knows
that this is his practice, the corresponding probabilities are
0.444,0.121, 0.435.
(3) If the punter stands pat and the banker thinks the
punter would usually ask for another card under the circumstances,
the probabilities are 0.490, 0.095, 0.415.
(4) If the punter asks for a card and the banker believes
he would usually stand pat with a 5, the probabilities are
0.447, 0.127, 0.426.
Could we calculate the advantage of disguising our
thoughts?