SOLUTION: A poker hand consists of five cards randomly dealt from a standard deck of 52 cards. The order of the cards does not matter. Determine the following probabilities for a 5-card poke

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Question 1056986: A poker hand consists of five cards randomly dealt from a standard deck of 52 cards. The order of the cards does not matter. Determine the following probabilities for a 5-card poker hand. Write your answers in percent form, rounded to 4 decimal places.
Determine the probability that all five of these cards are Spades.
Determine the probability that exactly 3 of these cards are face cards.

Answer by solve_for_x(190) About Me  (Show Source):
You can put this solution on YOUR website!
The number of ways that a hand of 5 spades can be drawn is:

C(13, 5) * C(39, 0) = %2813%21%2F%285%21%2813-5%29%21%29%29+%2A+%2839%21%2F%280%21%2839-0%29%21%29%29 = 1287 * 1 = 1287 ways

The number of ways that ANY 5-card hand can be drawn from a deck of 52 cards is:

C(52, 5) = 52%21+%2F+%285%21%2852-5%29%21%29 = 2598960

The probability is then:

P(5 spades) = 1287%2F2598960 = 0.00049520 = 0.0495%

For the hand with exactly 3 face cards, there are 12 face cards in the deck (3 in each suit),
and 52 - 12 = 40 non-face cards.

The number of ways of drawing a 5-card hand with exactly 3 face cards is equal to
the number of ways of selecting 3 out of 12 face cards, multiplied by the number of
ways of selecting 2 of the 40 non-face cards:

C(12, 3) * C(40, 2) = %2812%21%2F%283%21%2812-3%29%21%29%29+%2A+%2840%21%2F%282%21%2840-2%29%21%29%29 = 220 * 780 = 171600

The number of ways of drawing any 5-card hand is still 2598960 (from above)

The probability is then:

P(exactly 3 face cards) = 171600%2F2598960 = 0.0660264 = 6.6026%