A deck of cards contains 52 cards, divided equally among four suits, Spades (S), Clubs (C), Diamonds (D), and Hearts (H). Each suit has 13 cards which are designated: \((2,3,4,5,6,7,8,9\), \(10, \mathrm{~J}, \mathrm{Q}, \mathrm{K}, \mathrm{A}\) ). Assume that the deck is always well shuffled so it is equally likely to receive any card in the deck, when a card is dealt. (a) If a dealt hand consists of five cards, how many different hands can one be dealt (assume the cards in the hand can be received in any order)? (b) If the game is poker, what is the probability of being dealt a Royal Flush (10, J, Q, K, and A all of one suit)? (c) If one is dealt a hand with seven cards, and the first four cards are spades, what is the probability of receiving, at least one additional spade?

Step by step solution

01

Calculate the total number of 5-card hands

A deck contains 52 cards. To determine how many different 5-card hands can be dealt, use the combination formula \(\binom{n}{k}\), where \ is the total number of cards (52) and \k\ is the number of cards in the hand (5). The formula is \(\binom{52}{5}\): \[\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!\cdot47!}\]

02

Simplify the combination

Simplify \(\binom{52}{5}\) using factorials: \[\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2598960\]

03

Interpret the result for part (a)

The total number of different 5-card hands that can be dealt from a standard 52-card deck is 2598960.

04

Define a Royal Flush

A Royal Flush consists of the cards 10, J, Q, K, and A all in one suit (Spades, Clubs, Diamonds, or Hearts).

05

Calculate the probability of a Royal Flush

There are 4 possible Royal Flushes (one for each suit). The probability is the number of favorable outcomes (4 Royal Flushes) divided by the total number of 5-card hands (2598960): \[\frac{4}{2598960} = \frac{1}{649740}\]

06

Calculate the number of desired outcomes for part (c)

If the first four cards are spades and there are 13 spades in total, there are 9 remaining spades in the deck. Of the 48 remaining cards (52 - 4), 9 are spades, and 39 are not spades.

07

Calculate the probability of receiving at least one additional spade

Calculate the number of ways to get 3 non-spades out of the remaining 48 cards and the total number of ways to get 3 cards: \[\text{P(no spade in the next 3 cards)} = \frac{\binom{39}{3}}{\binom{48}{3}}\] The probability of getting at least one spade is therefore \[1 - \text{P(no spade)} = 1 - \frac{\binom{39}{3}}{\binom{48}{3}}\]

08

Simplify the probability for at least one additional spade

Simplify using the combination formula: \[\binom{39}{3} = \frac{39 \times 38 \times 37}{3 \times 2 \times 1} = 9139\] \[\binom{48}{3} = \frac{48 \times 47 \times 46}{3 \times 2 \times 1} = 17296\] Then, \[\text{P(no spade in next 3 cards)} = \frac{9139}{17296} \Rightarrow \text{P(at least one spade)} = 1 - \frac{9139}{17296} = \frac{8157}{17296}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arrangement, and combination of objects. In card games, combinatorics helps us figure out how many possible hands can be dealt or how likely certain hands are.
For example, when calculating the number of 5-card hands from a 52-card deck, we use the combination formula \(\binom{n}{k}\). This formula helps us find out how many ways we can choose \({k}\) items from \({n}\) items without regard to the order of selection.
Let's break down the combination formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, \({n}\) is the total number of items, \({k}\) is the number of items to choose, and \(!\) denotes the factorial operation, meaning the product of all positive integers up to the number.

Royal Flush

A Royal Flush is the highest-ranking hand in poker. It consists of the cards 10, J, Q, K, and A all in the same suit. There are four possible Royal Flushes in a standard deck, one for each suit: Spades, Clubs, Diamonds, and Hearts.
Because of its rarity and value, understanding the probability of being dealt a Royal Flush can be intriguing. Imagine you are playing poker, and you are hoping for that elusive combination. Knowing how to calculate this probability helps you appreciate just how special a Royal Flush really is.

Probability Calculation

In probability calculation, we deal with the likelihood of an event occurring. For example, to find the probability of being dealt a Royal Flush, we calculate the ratio of favorable outcomes to the total number of possible 5-card hands.
In this case, there are 4 favorable outcomes (one Royal Flush per suit) out of 2,598,960 possible 5-card hands. Therefore, the probability is calculated as:\br \(\frac{4}{2598960} = \frac{1}{649740}\).
Applying similar techniques, we can calculate other probabilities, such as the chance of getting an additional spade when dealt a 7-card hand.

Binomial Coefficient

The binomial coefficient is instrumental in combinatorics for finding the number of ways to choose \({k}\) items from \({n}\) items. It is represented as \(\binom{n}{k}\).
When calculating the probability of an event, the binomial coefficient often comes into play. For example, when calculating the number of ways to get 3 non-spades out of the remaining 48 cards, we use the binomial coefficient as follows: \(\binom{39}{3} = \frac{39 \times 38 \times 37}{3 \times 2 \times 1} = 9139\).
This is an essential part of determining the probability of various card hand combinations.