91Ó°ÊÓ

You are dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52-card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

Step by step solution

01

Calculate Total Ways to Select 5 Cards

Determine how many ways 5 cards can be chosen from a 52-card deck using the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n = 52 \) and \( r = 5 \). Compute \( C(52, 5) \).

02

Calculate Ways to Select Three of the Same Card

Each of the 13 ranks (twos, threes, etc.) has 4 cards. Use the combination formula to determine how many ways 3 cards can be chosen from these 4 cards: \[ C(4, 3) = \frac{4!}{3!(4-3)!} \] Since this can be any of the 13 ranks, multiply the result by 13.

03

Determine Ways to Select the Remaining 2 Cards

After selecting three of a kind, there are 12 remaining ranks. The combination formula can be used to find how to choose 2 ranks out of these 12 remaining ranks: \[ C(12, 2) \] Since any of the 4 cards can be chosen for each rank, multiply by \( 4 \times 4 \). Compute the total number of combinations.

04

Use Multiplication Rule to Calculate Total Ways for the Specific Hand

Multiply the number of ways to select three of a kind by the number of ways to select the remaining 2 cards. \[ \text{Total ways} = 13 \times C(4, 3) \times C(12, 2) \times 4^2 \]

05

Compute the Probability

The probability of being dealt the three of a kind is the ratio of the specific hand combinations to the total ways of selecting 5 cards from a 52-card deck: \[ P(\text{Three of a kind}) = \frac{\text{Total ways of specific hand}}{C(52, 5)} \]

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Key Concepts

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

Combination Formula

The combination formula is essential in determining the number of ways to choose a subset of items from a larger set without regard to the order of selection. It's represented as \(C(n, r) = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose.
For example, in poker, when determining how many ways we can select 5 cards from a 52-card deck, we apply this combination formula with \(n = 52\) and \(r = 5\).
This simplifies our problem and provides a systematic approach to calculating probabilities.

Card Probability

Card probability involves calculating the likelihood of drawing specific cards from a deck. Each event must be analyzed to understand the chances.
For instance, in determining how many ways to get three of a kind in poker:

• There are 13 ranks (twos, threes, etc.), each with 4 cards.
• We use the combination formula \(C(4, 3)\) to find out how many ways we can pick 3 out of these 4 cards, which equals 4 ways.
• Explicitly for three of a kind, we then multiply by the number of ranks available, 13.

This forms the basic understanding of calculating the probability of certain hands in poker.

General Multiplication Rule

The General Multiplication Rule helps in finding the probability of a series of independent events occurring together. To use this rule in poker probabilities:

• First, find the number of ways to pick 3 similar cards from a specific rank.
• Next, calculate the ways to pick remaining 2 cards from the surviving ranks.
• Finally, use the multiplication rule by multiplying these different ways together to understand the total combinations of the desired events.

This method helps in breaking down complex problems into simpler parts.

Poker Hands Probability

Poker hands probability focuses on calculating the chances of drawing specific hand combinations. In our three of a kind example, we can summarize the different steps needed to find the correct probability:

• Calculate the total number of ways to choose 5 cards from 52 using \(C(52, 5)\).
• Find the number of ways to select 3 same cards from any rank using \(13 \times C(4, 3)\).
• Determine the ways to pick the remaining 2 cards from different ranks using \(C(12, 2) \times 4 \times 4\).
• Combine these steps using the General Multiplication Rule.
• Finally, divide the result by the total combinations possible (\(C(52, 5)\)) to get the probability.

This structured approach makes it easier to tackle and understand poker probabilities.