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A card is drawn from a pack of 52 cards and a gambler bets that it is a spade or an ace. Which one of the following are the odds against his winning this bet?
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The correct answer is:
9 to 4
Probability of getting a spade $=\frac{13}{52}$
Probability of an ace $=\frac{4}{52}$
and probability of getting a spade ace $=\frac{1}{52}$
$\therefore \quad$ Required probability $=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$
Odds against his winning $=\frac{1-\frac{4}{13}}{\frac{4}{13}}=\frac{\frac{9}{13}}{\frac{4}{13}}=\frac{9}{4}$
Probability of an ace $=\frac{4}{52}$
and probability of getting a spade ace $=\frac{1}{52}$
$\therefore \quad$ Required probability $=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$
Odds against his winning $=\frac{1-\frac{4}{13}}{\frac{4}{13}}=\frac{\frac{9}{13}}{\frac{4}{13}}=\frac{9}{4}$
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