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A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 through 80. A player can select from 1 to 15 numbers; a win occurs if some fraction of the player's chosen subset matches any of the 20 numbers drawn by the house. The payoff is a function of the number of elements in the player's selection and the number of matches. For instance, if the player selects only 1 number, then he or she wins if this number is among the set of \(20,\) and the payoff is \(\$ 2.20\) won for every dollar bet. (As the player's probability of winning in this case is \(\frac{1}{4},\) it is clear that the "fair" payoff should be \(\$ 3\) won for every \(\$ 1\) bet.) When the player selects 2 numbers, a payoff (of odds) of \(\$ 12\) won for every \(\$ 1\) bet is made when both numbers are among the \(20 .\) (a) What would be the fair payoff in this case? Let \(P_{n, k}\) denote the probability that exactly \(k\) of the \(n\) numbers chosen by the player are among the 20 selected by the house. (b) Compute \(P_{n, k}\) (c) The most typical wager at Keno consists of selecting 10 numbers. For such a bet, the casino pays off as shown in the following table. Compute the expected payoff:

Step by step solution

01

Calculate the probability of a win with 2 numbers

The probability of a player's two numbers both being drawn by the house is the total number of ways for the house to draw both of the player's chosen numbers divided by the total number of ways for the house to draw any 20 numbers. This is given by: \(P(\text{win}) = \frac{\dbinom{2}{2}\dbinom{78}{18}}{\dbinom{80}{20}}\)

02

Compute the fair payoff

A fair payoff would be the inverse of the winning probability, multiplied by $1 to get the dollar amount that should be won per dollar bet: \(\text{fair payoff} = \frac{1}{P(\text{win})}\) Now, we can calculate the fair payoff. #Step 2: Compute \(P_{n, k}\), the probability of exactly k matches out of n chosen numbers# We are given the formula \(P_{n, k}\), and we need to compute the probability.

03

Derive the formula for \(P_{n, k}\)

To find the probability of exactly k matches out of n chosen numbers, we can apply a similar logic as in Step 1. The total number of ways to get k matching numbers and n-k non-matching numbers is: \(\dbinom{n}{k}\dbinom{80-n}{20-k}\) The total number of ways for the house to draw 20 numbers from 80 is: \(\dbinom{80}{20}\) Then, the probability of getting exactly k matches out of n chosen numbers is: \(P_{n, k} = \frac{\dbinom{n}{k}\dbinom{80-n}{20-k}}{\dbinom{80}{20}}\) #Step 3: Compute the expected payoff for selecting 10 numbers# We are given a table of the payoffs for each number of matches for a 10-number wager. We need to calculate the expected payoff.

04

Calculate the expected payoff

For each case, we will multiply the probability of getting the specified number of matches (using the formula for \(P_{10, k}\)) by the corresponding payoff. Then, we will sum all these values. Expected payoff = \(P_{10, 0}\) x (loss) + \(P_{10, 5}\) x (payoff for 5 matches) + ... + \(P_{10, 10}\) x (payoff for 10 matches) Now that we have found the steps for solving each part of the problem, we can calculate the values needed to answer the exercise.

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Key Concepts

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

Combinatorics

Combinatorics is all about counting the number of ways things can happen. In our Keno example, we're interested in exploring how the numbers are drawn. By using mathematical techniques, we can evaluate how many combinations of numbers exist. The math behind combinatorics often involves concepts like permutations and combinations. For instance, when dealing with Keno, you might use combinations to figure out how many ways you can choose 20 numbers from a pool of 80.

One of the key tools in combinatorics is the combination formula, which is \[\dbinom{n}{r} = \frac{n!}{r!(n-r)!}\]Here, \(n!\) ("n-factorial") represents the product of all positive integers up to \(n\), and this formula tells us how many ways we can choose \(r\) numbers from a set of \(n\) numbers.

• Permutations - where order matters, such as the arrangement of certain numbers.
• Combinations - where order does not matter, relevant in Keno where only the choice of numbers matters.

Understanding combinations in Keno helps calculate probabilities, such as the chance of a player selecting the correct numbers or determining how likely it is for an event to occur.

Expected Value

The expected value is like predicting the average outcome of an uncertain event. When playing Keno or any other gambling game, expected value gives you an idea of how much you could expect to win or lose per game, on average. It's a critical concept in probability and gambling mathematics.

To find the expected value, consider each possible outcome and multiply its value by the probability it'll happen. Then, sum these results. For Keno, you'd factor in all possible scenarios where you hit a certain number of matches and multiply those by the corresponding payoff. Through summing these, you obtain the expected payoff of the game. This concept allows players to make informed decisions about gambling and assess whether a game is in their favor.

• Helps determine fair payouts, indicating how much one should expect to win in a fair setting.
• Guides players in strategic decision-making.

Expected value is central to understanding whether a Keno game offers a fair chance to win or if the odds favor the casino.

Binomial Coefficient

The binomial coefficient plays a vital role in calculating probabilities, especially in games of chance like Keno. It is represented by \(\dbinom{n}{k}\) and is used to compute the number of ways to select \(k\) successes (like number matches) from \(n\) trials (chosen numbers). In our Keno context, the binomial coefficient helps calculate the probability of drawing exactly \(k\) numbers correctly.

• The formula \(\dbinom{n}{k} = \frac{n!}{k!(n-k)!}\) is applied.
• It's fundamental to determining the probability distribution across possible outcomes in a gambling scenario.

In this context, \(\dbinom{10}{k}\) might represent the player's chosen numbers in Keno when calculating probabilities for getting the correct matches against the 20 numbers drawn by the casino. Understanding the binomial coefficient allows players to calculate their winning probabilities accurately and decide how many numbers to bet in order to maximize their chances of winning.

Gambling Mathematics

Gambling mathematics involves the rigorous use of mathematical concepts to analyze games of chance like Keno. It combines probability, statistics, and other mathematical tools to calculate odds, potential payouts, and strategies.

In Keno, for example:

• Probability helps determine how likely you are to win based on the numbers chosen and the numbers drawn by the casino.
• Expected value helps assess the average payoff from playing multiple games, informing whether a game's setup is lucrative or not.
• Combinatorics accounts for the number of possible ways numbers can be drawn and arranged.

Overall, gambling mathematics provides players with insights that allow them to make smarter betting decisions. It informs players whether certain games are worth playing based on a calculated risk versus reward. For players in Keno, understanding these principles can make the difference between blindly gambling and placing informed, strategic bets.