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

A standard deck has 52 cards, of which 13 are spades. The probability that the first card drawn is a spade is given by the ratio of spades to the total number of cards. Therefore, the probability is \( \frac{13}{52} \).

02

Conditional Probability for Second Card

Having drawn a spade first, 12 spades remain out of 51 cards. The probability that the second card drawn is a spade, given that the first card was a spade, is \( \frac{12}{51} \).

03

Conditional Probability for Third Card

With two spades drawn, 11 spades remain out of 50 cards. The probability for the third card to be a spade is \( \frac{11}{50} \).

04

Conditional Probability for Fourth Card

After drawing three spades, 10 spades remain out of 49 cards. The probability for the fourth card, given that the first three are spades, is \( \frac{10}{49} \).

05

Conditional Probability for Fifth Card

After four spades are drawn, 9 spades remain out of 48 cards. The probability of drawing a spade as the fifth card is \( \frac{9}{48} \).

06

Probability of a Flush in Spades

The probability of drawing five spades in a row is the product of the individual probabilities: \[ \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} \times \frac{10}{49} \times \frac{9}{48}. \] Calculating this product gives the probability of a spade flush.

07

Probability of a Flush in any Suit

Since the probability of drawing five cards of the same suit is the same for any suit, multiply the probability of a flush in spades by 4 (one for each suit). Thus, the probability of a flush in any suit is: \[ 4 \times \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} \times \frac{10}{49} \times \frac{9}{48}. \]

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 the probability of an event occurring, given that another event has already occurred. In other words, it's how likely something is to happen, knowing that something else has already happened. This is a key concept in probability theory, especially when dealing with sequential events like drawing cards from a deck.

For instance, in the context of drawing cards, once a spade is drawn from a deck, the deck changes. You use conditional probability to determine the likelihood of drawing another spade, knowing that one has been removed. The adjustment of probabilities is continuous and depends on prior events, which reduces both the number of total cards and the number of spades available.

• The probability the first card is a spade is simply the number of spades out of the total cards: \(\frac{13}{52}\).
• Once a spade is drawn, only 12 spades remain from 51 cards: \(\frac{12}{51}\).
• This pattern continues; each step further confirms how important it is to factor in previous draws.

Understanding conditional probability is crucial to solving this problem and many others in probability."

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It's often used to solve problems related to probability, as seen with card games like poker. When calculating the probability of a flush in poker, combinatorial methods help us understand how likely specific outcomes are.

In this problem, we're using combinatorics to understand the sequence of drawing spades. With each card drawn, the total number of possible combinations changes. Here's why:

• Initially, you can draw any of the 13 spades from 52 cards.
• After one spade is drawn, only 12 spades remain with 51 cards total, changing the combinations possible.
• Combinatorics helps explain the sequential probability calculations, as it accounts for these changing conditions.

This approach lends itself perfectly to probabilistic problems by logically reducing the available outcomes with each step.

Statistical Reasoning

Statistical reasoning allows us to make sense of data and uncertain events. It helps in predicting the likelihood of different outcomes. With the problem of finding the probability of a flush, statistical reasoning provides a clear, logical approach to probability and combinatorial analysis.

Here's how you apply statistical reasoning:

• Consider the number of successful outcomes versus the total possible outcomes.
• Factor in conditional probabilities; with each spade drawn, reassess probabilities as the conditions change.
• Multiply probabilities to account for sequential events, such as drawing five spades in succession.

The ability to predict and understand the probability of a flush in poker is a practical application of statistical reasoning. By applying this reasoning, you logically work through the problem to find the likelihood of drawing five cards of the same suit in sequence. It’s an essential skill not only in games but in analyzing similar probabilistic problems in various fields.