Question

66 cards are drawn from a standard deck without replacement. What is the probability that at least one of the cards drawn is a spade? Express your answer as a fraction or a decimal number rounded to four decimal places.

Answer 1

Answer:

0.8397

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

The order in which the cards are chosen is not important. Also, they are drawn without replacement. So we use the combinations formula to solve this question.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

What is the probability that at least one of the cards drawn is a spade?

Either none is a spade, or at least one is a spade. The sum of the probabilities of these outcomes is 1.

The standard deck has 52 cards, of which 13 are spades. So

Probability that none are spades:

Desired outcomes:

6 cards from a set of 52 - 13 = 39. So

$D = C_{39,6} = \frac{39!}{6!33!} = 3262623$

Total outcomes:

6 cards from a set of 52. So

$T = C_{52,6} = \frac{52!}{6!46!} = 20358520$

Probability:

$p = \frac{D}{T} = \frac{3262623}{20358520} = 0.1603$

Probability that at least one is a spade:

1 - 0.1603 = 0.8397

The answer is 0.8397