QuestionCalculate the number of diamond flushes in a 3-card poker hand and its probability. Also, find total 3-card hands.

Studdy Solution

STEP 1

Assumptions1. A standard deck of52 cards is used. . A diamond flush is a3-card hand consisting of all diamond cards.
3. There are13 diamond cards in a deck.
4. The order of the cards does not matter.

STEP 2

First, we need to find the total number of possible-card poker hands. This can be found using the combination formula, which is given by $C(n, k) = \frac{n!}{k!(n-k)!}$ where- $n$ is the total number of items, - $k$ is the number of items to choose, - $C(n, k)$ is the number of combinations of $n$ items taken $k$ at a time, - $!$ denotes factorial.

STEP 3

Plug in the values for $n$ and $k$ to calculate the total number of possible3-card poker hands.
$C(52,3) = \frac{52!}{3!(52-3)!}$

STEP 4

Calculate the total number of possible3-card poker hands.
$C(52,3) = \frac{52!}{3!(49)!}$

STEP 5

implify the factorial expressions and calculate the total number of possible3-card poker hands.
$C(52,3) = \frac{52 \times51 \times50}{3 \times2 \times1} =22,100$

STEP 6

Now, we need to find the number of possible diamond flushes. This can also be found using the combination formula. $C(13,3) = \frac{13!}{3!(13-3)!}$

STEP 7

Calculate the number of possible diamond flushes.
$C(13,3) = \frac{13!}{3!(10)!}$

STEP 8

implify the factorial expressions and calculate the number of possible diamond flushes.
$C(13,3) = \frac{13 \times12 \times11}{3 \times2 \times1} =286$

STEP 9

Finally, we need to find the probability of being dealt a diamond flush. This can be found by dividing the number of possible diamond flushes by the total number of possible3-card poker hands.
$(Diamond\, Flush) = \frac{Number\, of\, Diamond\, Flushes}{Total\, Number\, of\,3-card\, Poker\, Hands}$

STEP 10

Plug in the values for the number of diamond flushes and the total number of3-card poker hands to calculate the probability.
$(Diamond\, Flush) = \frac{286}{22,100}$

STEP 11

Calculate the probability of being dealt a diamond flush.
$(Diamond\, Flush) = \frac{286}{22,100} \approx0.01294$ The number of possible diamond flushes in a3-card poker hand is286, and the probability of being dealt a diamond flush is approximately0.01294 or.294%.

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