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Chapter 5: Problem 129

Consider the Uses and Misuses section in this chapter on poker, where we learned how to calculate the probabilities of specific poker hands. Find the probability of being dealt a. three of a kind b. two pairs \(\mathrm{c}\). one pair

Short Answer

Expert verified
The probabilities of being dealt a three of a kind, two pairs and one pair in a poker game are approximately 0.021128, 0.047539 and 0.422569 respectively.

Step by step solution

01

Combination Calculation for Three of a Kind

Three of a kind means having three cards of the same face (eg. three Queens, three twos etc.) and two cards of any other faces, which will be different from each other. The calculation will be as follows: \n\n Choose one face for three-of-a-kind: \(13C1 = 13\). \n\n Choose 3 cards from the 4 of this face: \(4C3 = 4\). \n\n Then choose 2 different faces for the other two cards: \(12C2 = 66\). \n\n Finally, choose 1 card from each of the 2 chosen faces: \((4C1)^2 = 16\). \n\n Multiply all of these values together: \(13 * 4 * 66 * 16 = 54,912\) combinations.
02

Combination Calculation for Two Pairs

Two pairs means two cards of a face and two cards of another face and one card of another face. The calculation will be: \n\n We choose 2 faces for the pairs: \(13C2 = 78\). \n\n Then, we choose 2 cards from each chosen face \( (4C2)^2 = 36\). \n\n We choose a face for the remaining card: \(11C1 = 11\). \n\n Finally, we choose a card from this face: \(4C1 = 4\). \n\n Multiply all these values together: \(78 * 36 * 11 * 4 = 123,552\) combinations.
03

Combination Calculation for One Pair

One pair means two cards of a face and the other three cards are of different faces. The calculation will be: \n\n Choose one face for the pair: \(13C1 = 13\). \n\n Choose 2 cards from this face: \(4C2 = 6\). \n\n We choose the 3 faces for the remaining cards: \(12C3 = 220\). \n\n We choose one card from each of the chosen faces: \((4C1)^3 = 64\). \n\n Multiply all of these values together, \(13 * 6 * 220 * 64 = 1,098,240\) combinations.
04

Calculate Probability

The total possible outcomes when dealing 5 cards from a standard deck is \(52C5 = 2,598,960\). \n Thus the Probability of each hand is: \n - Three of a kind: \(54912 / 2598960 = 0.021128\). \n - Two Pair: \(123552 / 2598960 = 0.047539\). \n - One Pair: \(1098240 / 2598960 = 0.422569\).

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Key Concepts

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

Combinatorics
When it comes to calculating the probabilities of various poker hands, combinatorics plays a crucial role. Combinatorics is a branch of mathematics focused on counting, selection, and arrangement of objects. In poker, combinatorics helps determine how many ways we can form different hands from a deck of cards. Here's how this is done:
  • A standard deck has 52 cards, and in most poker games, 5 cards make a hand.
  • To calculate possibilities for hands like "three of a kind," "two pairs," or "one pair," we use combinations.
  • The term \(nCk\) or the combination formula is used, which computes the number of ways to select \(k\) items from a set of \(n\) without regard to order.
  • For example, \(13C1\) is used to choose the face value for a pair or three of a kind.
Combinatorics simplifies the otherwise daunting task of evaluating finite, structured possibilities in card games like poker. By systematically using combination formulas, we uncover insights on how likely different poker hands are.
Poker hands
Poker hands are groupings of cards in poker games that determine the outcome based on pre-established rules. Creating winning poker hands is a mix of skill and luck, heavily dependent on the randomness of card dealing but also on the player's strategic moves.
  • "Three of a Kind" occurs when a player holds three cards of the same rank, along with two unrelated cards.
  • "Two Pairs" involves two different pairs and a fifth card of a separate rank.
  • "One Pair" is when there are two cards of a similar rank and three other unrelated cards, all of distinct ranks.
Each hand type has a specific probability of occurring, calculated using combinatorics. Understanding the different poker hands and their probabilities is key in making informed decisions during the game. It allows players to assess their odds and adjust their strategies accordingly.
Statistics education
Statistics education involves learning about data collection, analysis, interpretation, and presentation. One application of statistics is in understanding probabilities, which are vital in games like poker. By studying statistics, students learn how to make sense of uncertain events.
  • Probability is a measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).
  • For poker hands, probability is determined by dividing the number of favorable outcomes by the total possible outcomes.
  • Hands like "one pair" have different probabilities compared to "three of a kind" or "two pairs," illustrating the statistical concepts of relative frequency.
Through the lens of poker, students can grasp core statistical concepts effortlessly. It provides a practical example of how probability and combinations interact, encouraging analytical thinking and enhancing risk assessment skills necessary not just in card games, but in real-life scenarios.

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Most popular questions from this chapter

An instant lottery ticket costs \(\$ 2\). Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of \(\$ 5\) each, 100 tickets have a prize of \(\$ 10\) each, 5 tickets have a prize of \(\$ 1000\) each, and 1 ticket has a prize of \(\$ 5000 .\) Let \(x\) be the random variable that denotes the net amount a player wins by playing this lottery. Write the probability distribution of \(x\). Determine the mean and standard deviation of \(x\). How will you interpret the values of the mean and standard deviation of \(x ?\)

Bender Electronics buys keyboards for its computers from another company. The keyboards are received in shipments of 100 boxes, each box containing 20 keyboards. The quality control department at Bender Electronics first randomly selects one box from each shipment and then randomly selects 5 keyboards from that box. The shipment is accepted if not more than 1 of the 5 keyboards is defective. The quality control inspector at Bender Electronics selected a box from a recently received shipment of keyboards. Unknown to the inspector, this box contains 6 defective keyboards. a. What is the probability that this shipment will be accepted? b. What is the probability that this shipment will not be accepted?

An insurance salesperson sells an average of \(1.4\) policies per day. a. Using the Poisson formula, find the probability that this salesperson will sell no insurance policy on a certain day b. Let \(x\) denote the number of insurance policies that this salesperson will sell on a given day. Using the Poisson probabilities table, write the probability distribution of \(x\). c. Find the mean, variance, and standard deviation of the probability distribution developed in part b.

A household receives an average of \(1.7\) pieces of junk mail per day. Find the probability that this household will receive exactly 3 pieces of junk mail on a certain day. Use the Poisson probability distribution formula.

In a list of 15 households, 9 own homes and 6 do not own homes. Four households are randomly selected from these 15 households. Find the probability that the number of households in these 4 who own homes is a. exactly 3 b. at most \(\overline{1}\) c. exactly 4
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