Question

in 5-card poker, find the probability of being dealt the following hand. refer to the table. note that a standard deck of playing cards has 52 cards—4 suits (clubs, diamonds, hearts, spades), where each suit has 13 cards (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king). a diamonds flush (not a royal flush or a straight flush) the probability of being dealt a diamonds flush is (type an integer or decimal rounded to eight decimal places as needed.)

event e	number of favorable outcomes
straight flush	36
four of a kind	624
full house	3744
flush	5108
straight	10,200
three of a kind	54,912
two pairs	123,552
one pair	1,098,240
no pair	1,302,540
total	2,598,960

Explanation:

Step1: Identify favorable and total outcomes

A diamonds flush (not royal or straight flush) is a type of flush. From the table, total number of flushes is 5108, but we need only diamonds flush (not royal or straight). A royal flush in diamonds is 1 (since royal flush has 4 total, 1 per suit), and straight flush in diamonds: a straight flush has 10 possible sequences (A-5 to 10-A) per suit, so 10. But wait, the table's "Flush" event: total flushes are 5108, which includes all flushes (non-straight, non-royal). Wait, actually, the problem says "diamonds flush (not a royal flush or a straight flush)". So first, total number of 5 - card hands: \( \binom{52}{5}=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960 \) (matches the total in the table: 2598960). Now, for a diamonds flush (5 cards of diamonds, not royal or straight). Number of diamonds in a deck: 13. Number of ways to choose 5 diamonds: \( \binom{13}{5}=\frac{13!}{5!8!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287 \). Now, subtract the royal flush in diamonds (1: 10, J, Q, K, A) and the straight flushes in diamonds. A straight flush in diamonds: a sequence of 5 consecutive cards. The possible sequences in diamonds: A - 5, 2 - 6, 3 - 7, 4 - 8, 5 - 9, 6 - 10, 7 - J, 8 - Q, 9 - K, 10 - A. Wait, but royal flush is 10 - A, so straight flushes in diamonds: total straight flushes per suit: 10 (from A - 5 to 10 - A), but royal flush is one of them (10 - A). So straight flushes in diamonds (excluding royal): 9? Wait, no: the table has "Straight flush" with 36 (4 suits × 9? Wait, 4 suits × 10 straight flushes? No, 4 suits × 9 non - royal straight flushes? Wait, the table says "Straight flush" has 36, "Royal flush" has 4 (1 per suit). So total straight flushes (including royal) would be 36+4 = 40, which is 4 suits × 10 (since 10 sequences per suit: A - 5 to 10 - A). So per suit, straight flushes: 10 (including royal). So for diamonds, straight flushes: 10 (including royal). So diamonds flush (not royal or straight flush): number of ways to choose 5 diamonds \( \binom{13}{5} \) minus royal flush (1) minus straight flushes (10 - 1=9? Wait, no: royal flush is a type of straight flush (the highest one). So straight flushes in diamonds: 10 (A - 5, 2 - 6, ..., 10 - A), and royal flush is 10 - A. So diamonds flush (not royal or straight flush) = \( \binom{13}{5}-1 - 9=1287 - 10 = 1277 \)? Wait, no, the table's "Flush" event: number of flushes is 5108. Let's check: total flushes: number of ways to choose 5 cards of a suit: \( 4\times\binom{13}{5}=4\times1287 = 5148 \). But the table has "Flush" as 5108. The difference is because we subtract the straight flushes (including royal). Total straight flushes (including royal): 4 suits × 10 = 40. So 5148-40 = 5108, which matches the table. So "Flush" in the table is non - straight, non - royal flushes. So for diamonds, the number of diamonds flushes (non - straight, non - royal) is total flushes in diamonds minus straight flushes in diamonds (including royal). Total flushes in diamonds: \( \binom{13}{5}=1287 \). Straight flushes in diamonds: 10 (including royal). So diamonds flushes (non - straight, non - royal): 1287 - 10 = 1277? Wait, no: 5148 (total flushes across 4 suits) - 40 (straight flushes across 4 suits)=5108 (total flushes in table). So per suit, flushes (non - straight, non - royal): 5108/4 = 1277. Yes! Because 4 suits, so 5108 divided by 4 is 1277. So that's the number of favorable outcomes for a diamonds flush (not royal or straight flush): 1277. Total number of 5 - card hands: 2598960.

Step2: Calculate probability

Probability \( P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{1277}{2598960}\approx0.0004913 \) (wait, no: wait, 5108 is total flushes (all suits, non - straight, non - royal). So diamonds flush (non - straight, non - royal) is 5108/4 = 1277? Wait, 5108 divided by 4: 5108÷4 = 1277. Yes. So probability is \( \frac{1277}{2598960}\approx0.0004913 \)? Wait, no, wait: let's recalculate \( \binom{13}{5}=1287 \), subtract 10 (straight flushes in diamonds: 10 sequences: A - 5, 2 - 6, ..., 10 - A), so 1287 - 10 = 1277. Then 1277/2598960. Let's compute that: 1277÷2598960≈0.0004913. Wait, but let's check with the table. The total number of flushes is 5108, so probability of a flush (any suit, non - straight, non - royal) is 5108/2598960≈0.001965. Then a diamonds flush would be 1/4 of that, since there are 4 suits. So 0.001965/4≈0.000491. Yes, that matches.

Answer:

\( \approx0.0004913 \) (or more accurately, \( \frac{1277}{2598960}\approx0.000491 \))