91Ó°ÊÓ

The probability of a flush. A poker player holds a flush when all five cards in the hand belong to the same suit (clubs, diamonds, hearts, or spades). We will find the probability of a flush when five cards are drawn in succession from the top of the deck. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well shuffled, each card drawn is equally likely to be any of those that remain in the deck. (a) Concentrate on spades. What is the probability that the first card drawn is a spade? What is the conditional probability that the second card drawn is a spade, given that the first is a spade? (Hint: How many cards remain? How many of these are spades?) (b) Continue to count the remaining cards to find the conditional probabilities of a spade for the third, the fourth, and the fifth card drawn, given in each case that all previous cards are spades. (c) The probability of drawing five spades in succession from the top of the deck is the product of the five probabilities you have found. Why? What is this probability? (d) The probability of drawing five hearts or five diamonds or five clubs is the same as the probability of drawing five spades. What is the probability that the five cards drawn all belong to the same suit?

Step by step solution

01

Probability of First Card Being a Spade

To find the probability that the first card drawn is a spade, we recognize that there are 13 spades in a full deck of 52 cards. Therefore, the probability that the first card is a spade is \( \frac{13}{52} \), which simplifies to \( \frac{1}{4} \).

02

Conditional Probability of Second Card Being a Spade

Assuming the first card drawn is a spade, there are now 12 spades left in a deck of 51 cards. The probability that the second card is also a spade is \( \frac{12}{51} \).

03

Conditional Probability of Third Card Being a Spade

Assuming the first two cards are spades, only 11 spades remain in a deck of 50 cards. The probability that the third card is a spade is \( \frac{11}{50} \).

04

Conditional Probability of Fourth Card Being a Spade

Assuming the first three cards are spades, there are now 10 spades left in 49 cards. Thus, the probability that the fourth card is a spade is \( \frac{10}{49} \).

05

Conditional Probability of Fifth Card Being a Spade

Assuming the first four cards are spades, we now have 9 spades remaining in a total of 48 cards. The probability that the fifth card is a spade is \( \frac{9}{48} \), which simplifies to \( \frac{3}{16} \).

06

Calculate the Probability of Five Spades in Succession

The probability of drawing five spades in a row is the product of the individual probabilities. Therefore, this probability is:\[ \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} \times \frac{10}{49} \times \frac{9}{48} = \frac{33}{16660} \]

07

Probability of a Flush with Any Suit

Since the probability of drawing five hearts, clubs, or diamonds is the same as for spades, we multiply the probability by 4 (one for each suit):\[ 4 \times \frac{33}{16660} = \frac{132}{16660} = \frac{3}{760} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Conditional Probability

Conditional probability is an essential concept in probability theory and statistics. It measures the probability of an event given that another event has already occurred. In the context of the flush probability exercise, we calculate the probability of drawing a spade given that certain cards have already been drawn.
For instance, if the first card drawn is a spade, the probability of drawing a spade as the next card changes because now there are fewer spades and fewer cards overall. This change reflects the condition that the first card was a spade, which influences the likelihood of subsequent events.

• The formula for conditional probability is expressed as \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(A|B) \) is the probability of event A occurring given that B is true.
• This type of probability helps make informed calculations when previous outcomes affect the likelihood of future events.

Deck of Cards

A standard deck of cards consists of 52 playing cards divided into four suits: spades, hearts, diamonds, and clubs. Each suit contains 13 cards ranging from Ace through King.
This structure is crucial when calculating probabilities in card games or related exercises. Knowing the number of cards and how they are distributed helps in understanding the mechanics of probability calculations in such scenarios.

• In probability, every card has an equal chance of being drawn initially.
• With each draw, the deck's size diminishes, altering the probabilities.

Studying card-related probability helps grasp more complex concepts like permutations and combinations used in larger probability calculations.

Flush Probability

Flush probability specifically refers to the likelihood of drawing five cards from the deck all of the same suit. This is a common requirement in poker games where achieving a flush represents a high-ranking hand.
In probability terms, achieving a flush when drawing five cards means all cards must belong to one suit like spades, and must be calculated carefully.

• The probability requires multiplying the probabilities of sequentially drawing each card of the same suit without replacement.
• In the context of the exercise, we calculated it for spades first and then adapted the same probability for the other suits.
• Understanding this concept improves strategic thinking both in probability exercises and card games.

Suit in Poker

A suit in poker refers to one of the four categories of cards: spades, hearts, diamonds, and clubs. Each category has an equal number of cards, which is significant in calculating probability because each initial draw from the deck has the same chance of being any suit.
In poker, specific combinations or achievements like flushes depend on having a certain number of cards from the same suit. This is why understanding suits are so critical.

• Each suit gives a uniform structure to the deck, making calculations like flush probability possible.
• Strategizing in poker often involves predicting or calculating the likelihood of completing a hand, such as a flush, based on known cards and suits.

By appreciating how suits operate in poker and probability, one can enhance both mathematical and strategic capabilities.