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Chapter 5: Problem 4

Texas hold 'em In the popular Texas hold 'em variety of poker, players make their best five-card poker hand by combining the two cards they are dealt with three of five cards available to all players. You read in a book on poker that if you hold a pair (two cards of the same rank) in your hand, the probability of getting four of a kind is \(88 / 1000\). (a) Explain what this probability means. (b) If you play 1000 such hands, will you get four of a kind in exactly 88 of them? Explain.

Short Answer

Expert verified
(a) Probability means 88 times out of 1000. (b) No, it's an expectation, not an exact count.

Step by step solution

01

Understanding the Probability

The probability of getting four of a kind if you have a pair in your hand is given as \( \frac{88}{1000} \). This means that over a large number of games where you start with a pair, on average, you can expect to get four of a kind 88 times out of every 1000 games. It is a measure of likelihood, not a guarantee of occurrences.
02

Expectation vs. Exact Outcomes

In statistics, probabilities are used to calculate expected outcomes, not exact results for a finite number of trials. So, if you play 1000 hands, you expect to achieve four of a kind approximately 88 times, but you may not get exactly 88 occurrences. The actual number can vary due to random chance.

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Key Concepts

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

Texas hold 'em
Texas hold 'em is one of the most popular forms of poker, played both in casinos and online. In this game, players aim to make the best five-card poker hand possible. They do this by using two hole cards (the cards dealt to a player) combined with five community cards that are revealed over a series of rounds. From these seven cards, each player makes their best hand, often resulting in intense strategic play.

Key aspects that make Texas hold 'em exciting include:
  • Strategic betting and bluffing, as players must decide to bet, check, or fold based on the strength of their hand and potential improvement.
  • The importance of understanding table dynamics and the tendencies of other players, which can influence decision-making.
  • Making educated guesses on opponents' hands and potential outcomes based on limited information.
While the rules of Texas hold 'em are simple, mastering the game involves understanding probabilities and making decisions based on incomplete information.
Expected outcomes
In the realm of probability, expected outcomes help us anticipate what is likely to occur over a large number of trials. Let's consider an example in the context of poker: if you have a pair in your hand, the probability of improving this pair to four of a kind in Texas hold 'em is known to be \( \frac{88}{1000} \). This means that, on average, you might expect to see this scenario play out approximately 88 times over 1,000 hands.

Here is what this implies:
  • The calculation of expected outcomes allows players to understand the long-term probabilities of specific hands occurring.
  • It's crucial when making decisions about staying in a round or folding early in the hand.
  • Expected values hinge on probability and help players manage their decisions in light of possible gains or losses over time.
Although expected outcomes give an average over time, in any given short run, results can vary greatly due to the role of chance.
Statistical variation
Statistical variation refers to the fluctuation or spread observed in potential outcomes when conducting experiments or playing hands of poker. It's the reason why in 1,000 hands, you might not get exactly 88 four-of-a-kind hands, even though that's the expected probability. Instead, you might see more or fewer due to the inherent randomness and variability in poker.

Key points about statistical variation in context:
  • Even when probabilities are known, individual outcomes can differ from the expected outcome due to natural variability.
  • Understanding variation is crucial to managing expectations and making informed decisions in poker.
  • Large sample sizes (more hands played) tend to bring averages closer to the expected outcomes, but the variance might be significant in smaller samples.
Players need to be aware that statistical variation can cause streaks of wins or losses, and strategize accordingly with risk management and patience.

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Most popular questions from this chapter

Preparing for the GMAT A company that offers courses to prepare students for the Graduate Management Admission Test (GMAT) has the following information about its customers: \(20 \%\) are currently undergraduate students in business; \(15 \%\) are undergraduate students in other fields of study; \(60 \%\) are college graduates who are currently employed; and \(5 \%\) are college graduates who are not employed. Choose a customer at random. (a) What's the probability that the customer is currently an undergraduate? Which rule of probability did you use to find the answer? (b) What's the probability that the customer is not an undergraduate business student? Which rule of probability did you use to find the answer?

You read in a book about bridge that the probability that each of the four players is dealt exactly one ace is about \(0.11 .\) This means that (a) in every 100 bridge deals, each player has one ace exactly 11 times. (b) in 1 million bridge deals, the number of deals on which each player has one ace will be exactly 110,000 . (c) in a very large number of bridge deals, the percent of deals on which each player has one ace will be very close to \(11 \%\) (d) in a very large number of bridge deals, the average number of aces in a hand will be very close to \(0.11 .\) (e) If each player gets an ace in only 2 of the first 50 deals, then each player should get an ace in more than \(11 \%\) of the next 50 deals.

Mac or \(\mathrm{PC}\) ? A recent census at a major university revealed that \(40 \%\) of its students mainly used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, \(67 \%\) of the school's students were undergraduates. The rest were graduate students. In the census, \(23 \%\) of respondents were graduate students who said that they used \(\mathrm{PCs}\) as their main computers. Suppose we select a student at random from among those who were part of the census. (a) Make a two-way table for this chance process. (b) Construct a Venn diagram to represent this setting. (c) Consider the event that the randomly selected student is a graduate student and uses a Mac. Write this event in symbolic form based on your Venn diagram in part (b). (d) Find the probability of the event described in part (c). Explain your method.

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, \(50 \%\) of people aged 13 and older in the United States are addicted to texting. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to texting, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to texting; a black card means he isn't. Continue until you have drawn 20 cards (without replacement) for the sample. (b) A tennis player gets \(95 \%\) of his serves in play during practice (that is, the ball doesn't go out of bounds). To simulate the player hitting 5 serves, look at 5 pairs of digits going across a row in Table \(D .\) If the number is between 00 and 94 , the serve is in; numbers between 95 and 99 indicate that the serve is out.

Urban voters The voters in a large city are \(40 \%\) white, \(40 \%\) black, and \(20 \%\) Hispanic. (Hispanics may be of any race in official statistics, but here we are speaking of political blocks. A mayoral candidate anticipates attracting \(30 \%\) of the white vote, \(90 \%\) of the black vote, and \(50 \%\) of the Hispanic vote. Suppose we select a voter at random. (a) Draw a tree diagram to represent this situation. (b) Find the probability that this voter votes for the mayoral candidate. Show your work. (c) Given that the chosen voter plans to vote for the candidate, find the probability that the voter is black. Show your work.
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