Problem 42 - Discrete Mathematics with Applications

Question

For the casino game football pools, a list of 10 football games is printed on a ticket. If one team is considered weaker than its opponent by the people who run the pool, that team is given enough points to make the game a tossup. Thus the probability of picking a winning team is 0.5. You pay \(\$ 1\) to play the game and select all 10 winners. If all your selections win, you get \(\$ 150 ;\) if nine win, you receive a consolation prize of \(\$ 20 ;\) otherwise, you lose your dollar. Compute your expected profit. (A. Sterrett, 1967 )

Step-by-Step Solution

Answer

The expected profit for the casino game football pools is \(-\$0.65\).

Step 1: Identify the binomial distribution parameters and the payouts for each outcome

Let X be a random variable representing the number of winning games the player picks. We will set up the problem as a binomial distribution with parameters n = 10 (number of games), and p = 0.5 (probability of picking the correct winner for a game). The payouts for each outcome are: 1. Picking 10 winners: $150 2. Picking 9 winners: $20 3. Picking fewer than 9 winners: \( -1 (losing the \)\$ 1$ paid to play) Now, we will calculate the probability of each outcome using the binomial distribution formula.

Step 2: Binomial probability formula

The binomial probability formula is: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) Where: - n is the number of trials (games) - k is the number of winning games - p is the probability of picking a winner for a single game For our problem: n = 10, and p = 0.5.

Step 3: Calculate the probabilities of each outcome

1. Probability of picking all 10 winners: \(P(X=10) = \binom{10}{10} (0.5)^{10} (1-0.5)^{0} = 0.000976\) 2. Probability of picking 9 winners: \(P(X=9) = \binom{10}{9} (0.5)^{9} (1-0.5)^{1} = 0.009766\) 3. Since we are looking for the expectation, we can consider the probability of picking fewer than 9 winners (8 or less) as the complementary probability. \(P(X<9) = 1 - P(X=10) - P(X=9) = 1 - 0.000976 - 0.009766 = 0.989258\)

Step 4: Multiply the probabilities with the respective payouts

1. Picking 10 winners: \(0.000976 * 150 = 0.1464\) 2. Picking 9 winners: \(0.009766 * 20 = 0.19532\) 3. Picking fewer than 9 winners: \(0.989258 * (-1) = -0.989258\)

Step 5: Compute the expected profit

The expected profit is the sum of the values from Step 4: Expected profit = \(0.1464 + 0.19532 - 0.989258 = -0.647538\) As promised, the expected profit is: \( -\$ 0.65\).

Key Concepts

Binomial Distribution

The binomial distribution is a cornerstone concept in probability and statistics, particularly when we deal with scenarios involving a fixed number of independent trials, each with two possible outcomes - success or failure. In the context of discrete mathematics, it helps us understand the likelihood of a specific number of successes in a given number of trials.

For any given trial, there's a constant probability of success (denoted as 'p') and failure (denoted as '1-p'), which leads us to the hallmark formula of the binomial distribution:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Here, \(n\) represents the number of trials, \(k\) is the number of successes, and \(\binom{n}{k}\) is a binomial coefficient representing the number of ways to choose \(k\) successes from \(n\) trials.

When calculating expected profit in gambling scenarios like the football pools game from our exercise, we use the binomial distribution to determine the probabilities for the various winning outcomes. Combining these probabilities with the potential payouts, which could be seen as payoffs for 'successes', allows us to estimate an expected monetary return over a long series of games - leading to a practical understanding of what could be gained (or lost) when participating regularly in such activities.

Probability

Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 indicates an impossible event, and 1 represents an event that is certain to happen. The core of probability lies in its axioms, which are the rules that relate to the likelihood of events.

Considering our exercise on the casino game, where the gamble is to pick winners in football games, probability helps us deduce the player's chances of winning based on the assumption that each game is a 'tossup'. This means that for each individual game, the probability \(p\) of picking the winning team is 0.5, or 50%. By applying the concepts of probability to the binomial distribution, we're able to quantify the expectations for different outcomes - ultimately allowing us to calculate the expected profit or loss from playing the game.

When the probabilities for all possible outcomes are known, as in the case of the binomial distribution, we can sum up the products of each outcome's value and its probability to find the expected value. In the realm of gambling, understanding these probabilities offers a sobering insight into the risks involved and the fairness of the game at hand.

Discrete Mathematics

Discrete mathematics involves the study of mathematical structures that are fundamentally discrete, rather than continuous. It includes topics such as logic, set theory, combinatorics, graph theory, and probability - with binomial distribution residing under its large umbrella.

In our casino game scenario, discrete mathematics provides the framework for analyzing finite, countable processes like a series of 10 football games. Calculating expected profit requires the use of concepts from discrete mathematics, specifically combinatorics in calculating binomial coefficients, and probability theory in evaluating the likelihood of distinct outcomes.

Moreover, discrete mathematics teaches us to think critically about the structure of problems. By dissecting the casino game into countable components (individual games) and applying binomial distribution, we adopt a systematic approach to understanding how each component (each game's outcome) contributes to the overall expectation (profit or loss). Such methodologies, indicative of discrete mathematics, are essential for solving complex problems in computer science, economics, and even to strategize games of chance, making it a highly practical and sought-after field of study.