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The probability of a flush A poker player holds a flush when all 5 cards in the hand belong to the same suit. We will find the probability of a flush when 5 cards are dealt. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck. (a) We will concentrate on spades. What is the probability that the first card dealt is a spade? What is the conditional probability that the second card is a spade given that the first is a spade? (b) Continue to count the remaining cards to find the conditional probabilities of a spade on the third, the fourth, and the fifth card given in each case that all previous cards are spades. (c) The probability of being dealt 5 spades is the product of the five probabilities you have found. Why? What is this probability? (d) The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades. What is the probability of being dealt a flush?

Step by step solution

01

Probability of the First Card

There are 52 cards in total, and 13 of them are spades. So, the probability of the first card being a spade is \( \frac{13}{52} \).

02

Conditional Probability of the Second Card

If the first card was a spade, then there are now 12 spades and 51 cards remaining. Thus, the conditional probability that the second card is a spade, given that the first was a spade, is \( \frac{12}{51} \).

03

Conditional Probability of the Third Card

After two spades have been dealt, there are 11 spades left and 50 cards remaining. Therefore, the conditional probability that the third card is a spade, given that the first two were spades, is \( \frac{11}{50} \).

04

Conditional Probability of the Fourth Card

If three spades have been dealt, there are 10 spades left and 49 cards remaining. Hence, the conditional probability that the fourth card is a spade, given that the first three were spades, is \( \frac{10}{49} \).

05

Conditional Probability of the Fifth Card

With four spades already dealt, 9 spades remain and 48 cards are left. So, the conditional probability that the fifth card is a spade, given that the first four were spades, is \( \frac{9}{48} \).

06

Calculating the Probability of Five Spades

The probability of being dealt 5 spades is the product of all these individual probabilities:\[\frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} \times \frac{10}{49} \times \frac{9}{48}\]This simplifies to \( \frac{33}{66640} \approx 0.000493 \), representing the probability of drawing 5 spades.

07

Probability of Being Dealt a Flush

Since there are four suits, the probability of being dealt a flush in any suit is four times the probability of a flush in spades. Thus, the probability of being dealt a flush is:\[4 \times \frac{33}{66640} = \frac{132}{66640} \approx 0.00198\]

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Key Concepts

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

Conditional probability

Conditional probability is a key concept in probability theory that helps us calculate the probability of an event, given that another event has already occurred. It answers the question: "What is the probability of event B, given that event A has happened?" For instance, in card games, it can be immensely helpful.
In our exercise, consider that you're playing poker and the first card dealt to you is a spade. There were originally 13 spades in a standard deck of 52 cards, making the probability of the first card being a spade \( \frac{13}{52} \).
Once the first spade is dealt, the deck changes. Now only 51 cards are left, and 12 of those are spades. This changes the probability that the second card is a spade to \( \frac{12}{51} \).
The process of calculating the probability of each successive card being a spade continues in this way, each time using the reduced deck, showcasing how conditional probability works elegantly and practically.

Poker probabilities

Poker probabilities refer to the chances of various hands and card combinations occurring in poker. Understanding these probabilities can significantly enhance a player's strategy and decision-making.
There are many different poker hands, each with their own unique probability of being dealt. A flush, like the one in our exercise, where all cards are of the same suit, has a specific probability that can be calculated using core probability principles.
The probability for each card being a part of a flush can be found by multiplying the conditional probabilities of each card being of the needed suit. For instance, the probability that all five cards dealt are spades was calculated as a sequential product of probabilities, revealing it to be a modest likelihood relative to other possible hands.

Probability of a flush

The probability of a flush in poker refers to the likelihood of having all five cards of the same suit in your hand. This probability can change depending on the variant of poker being played and the number of players involved.
In the case of our step-by-step solution, we saw that the probability of receiving five spades was calculated as \( \frac{33}{66640} \approx 0.000493 \). This is quite a rare occurrence, highlighting the flush's high value in poker hands.
However, since there are four suits, our total probability of receiving any flush (spades, hearts, diamonds, or clubs) is four times that of just spades. This was calculated as \( \frac{132}{66640} \approx 0.00198 \), still a relatively low probability, underscoring why flushes are distinguished as valuable during gameplay.

Card games probability

Card games probability involves analyzing the likelihood of drawing certain cards or combinations of cards from a shuffled deck. It forms the backbone of many strategic decisions in games like poker.
In our example, we calculated the probability of drawing a flush. This involved understanding not only the initial draw probabilities but also how each draw affects subsequent probabilities.
The beauty of card games lies in their complexity, where seemingly small improvements in understanding probability can offer significant strategic advantages. Predicting hands, calculating risks, and making informed decisions all rely on these fundamental principles, combining mathematical rigor with the art of betting.