Calculating Atomic Events in Five-Card Poker Hands

The question asks for the number of atomic events in the joint probability distribution of a five-card poker hand dealt from a standard deck of fifty-two cards. In probability, an atomic event refers to a single, specific outcome. In the context of dealing poker hands, each unique combination of five cards is a distinct atomic event.

To find the total number of possible five-card poker hands, we need to determine the number of ways to choose 5 cards from a set of 52 cards, where the order of selection does not matter. This is a classic problem in combinatorics and can be solved using the combination formula.

Understanding Combinations

A combination is a selection of items without regard to the order in which they are selected. The formula for the number of combinations of selecting \(k\) items from a set of \(n\) items is given by:

\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Where:

• \(n!\) is the factorial of \(n\), calculated as \(n \times (n-1) \times \dots \times 2 \times 1\).
• \(k!\) is the factorial of \(k\).
• \((n-k)!\) is the factorial of \(n-k\).

Applying the Combination Formula to Five-Card Poker Hands

In this problem:

• The total number of items to choose from, \(n\), is the number of cards in a standard deck, which is 52.
• The number of items to choose, \(k\), is the number of cards in a poker hand, which is 5.

So, we need to calculate the number of combinations of choosing 5 cards from 52, denoted as \(\binom{52}{5}\).

Using the formula:

\(\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!}\)

Expanding the factorials:

\(\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47!}{5 \times 4 \times 3 \times 2 \times 1 \times 47!}\)

We can cancel out the \(47!\) term from the numerator and the denominator:

\(\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1}\)

Now, we calculate the value:

Denominator: \(5 \times 4 \times 3 \times 2 \times 1 = 120\)

Numerator: \(52 \times 51 \times 50 \times 49 \times 48\)

Let's simplify the expression by dividing terms from the numerator by terms from the denominator:

\(\binom{52}{5} = \frac{52}{1} \times \frac{51}{3} \times \frac{50}{5 \times 2} \times \frac{49}{1} \times \frac{48}{4}\)

\(\binom{52}{5} = 52 \times 17 \times \frac{50}{10} \times 49 \times 12\)

\(\binom{52}{5} = 52 \times 17 \times 5 \times 49 \times 12\)

Calculating the product:

\(52 \times 17 = 884\)

\(5 \times 49 = 245\)

\(884 \times 245 \times 12\)

\(884 \times (245 \times 12)\)

\(245 \times 12 = 245 \times (10 + 2) = 2450 + 490 = 2940\)

\(884 \times 2940\)

Performing the multiplication:

\(884 \times 2940 = 2,598,960\)

Total Number of Atomic Events

The total number of possible five-card poker hands is 2,598,960. Each of these unique hands represents one atomic event in the joint probability distribution of five-card hands dealt from a standard deck.

Revision Table: Five-Card Hand Calculation

Additional Information: Probability and Poker

The total number of atomic events (the total number of possible hands) forms the denominator when calculating the probability of getting specific poker hands (like a Royal Flush, Four of a Kind, etc.). For example, to find the probability of a Royal Flush (which has only 4 possible specific combinations - one for each suit), you would divide the number of Royal Flushes (4) by the total number of possible hands (2,598,960).

Understanding combinations is fundamental in calculating probabilities in card games and other scenarios where the order of selection does not affect the outcome.