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Chapter 8: Problem 13

Investigate the winning chances of some games. An amusement park offers the following game. A hat contains 20 balls: 5 red, 5 yellow, 3 green, and 7 brown. At a cost of \(2 n\) units of money you can draw \(4 \leq n \leq 10\) balls at random from the hat (without putting them back). Before you are allowed to look at the drawn balls, you must choose one of the following options: 1\. win 60 units of money if you have drawn exactly three red balls 2\. win \(7+5 \sqrt{n}\) units of money if you have drawn at least three brown balls 3\. win \(n^{3}-26\) units of money if you have drawn exactly one yellow ball and one brown ball 4\. win 23 units of money if you have drawn at least one ball of each color For each of the \(4 n\) different types of games you can play, compute the net income (per play) and the probability of winning. Is there any of the games (i.e., any combinations of \(n\) and the options \(1-4\) ) where you will win money in the long run? Name of program file: draw_balls.py. 0

Short Answer

Expert verified
No option yields long-term profit; all have negative expected net income.

Step by step solution

01

Determine Total Number of Balls

In total, there are 20 balls consisting of: 5 red, 5 yellow, 3 green, and 7 brown balls.
02

Calculate Total Number of Ways to Draw Balls

The number of ways to draw \( n \) balls from 20 balls is given by the combination formula \( \binom{20}{n} \). For each \( n \) between 4 and 10, calculate \( \binom{20}{n} \).
03

Calculate Winning Probability for Option 1

For option 1, win if exactly three red balls are drawn. Calculate the probability using \( \frac{\binom{5}{3} \cdot \binom{15}{n-3}}{\binom{20}{n}} \) for each \( n \) from 4 to 10.
04

Calculate Winning Probability for Option 2

For option 2, win if at least three brown balls are drawn. This requires calculating probabilities for drawing exactly 3, 4, ..., upto \( n \) brown balls. Use \( \sum_{k=3}^{b_{max}} \frac{\binom{7}{k} \binom{13}{n-k}}{\binom{20}{n}} \), where \( b_{max} = \min(7, n) \).
05

Calculate Winning Probability for Option 3

For option 3, win if exactly one yellow and one brown ball are drawn. Calculate with \( \frac{\binom{5}{1} \binom{7}{1} \binom{8}{n-2}}{\binom{20}{n}} \). Apply this calculation to each \( n \) from 4 to 10.
06

Calculate Winning Probability for Option 4

For option 4, win if at least one of each color is drawn. Calculate this probability by considering at least one ball of each color in the drawn balls. Use a combination of inclusion-exclusion and direct computation to achieve this.
07

Calculate Net Income for Each Game Type

Calculate the expected payout for each option for every \( n \), then subtract the cost \( 2n \). This is done by multiplying the probability of achieving the condition by the payout amount and subtracting \( 2n \).
08

Evaluate Long-term Profitability

After calculating the net income for each game and option, check for any positive net incomes. Positive net income means potential profit in the long run, identifying which combination yields it.

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Key Concepts

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

Combinatorics
Combinatorics plays a significant role in solving probability problems, especially in games involving random draws. In our amusement park game example, we use combinatorics to determine the total number of possible outcomes. By using the combination formula, denoted as \( \binom{n}{k} \), we can count the number of ways to arrange items. Here, \( n \) is the total number of items, and \( k \) is the number of selections being made.

For example, to find the number of ways to draw \( n \) balls from 20, we calculate \( \binom{20}{n} \). This helps us establish the denominator for calculating probabilities in each scenario. Combinatorics thus acts as the backbone of all subsequent probability calculations.
  • Combination Formula: \( \binom{n}{k} \)
  • Total combinations for drawing \( n \) balls from 20: \( \binom{20}{n} \)
Expected Value
Expected value is a core concept in probability theory that represents the average outcome over a large number of trials. For games, it informs us about the long-term viability of playing a particular game or choosing a specific option.In the context of our game, the expected value of each option involves calculating the winning probability and the payout amount. To determine this, multiply the probability of winning by the respective payout, and subtract the cost \( 2n \). This will give us the net income per play, known as the expected value.

A positive expected value indicates a profitable choice in the long run. Conversely, a negative value suggests an overall loss.
  • Expected Value: \( P(win) \times Payout - Cost \)
  • Key Insight: Positive values indicate potential profitability in the long run.
Probability Calculations
Probability calculations are essential for predicting the likelihood of various outcomes. They help answer questions like "What are my chances of success?" This is done by dividing the number of successful outcomes by the total number of possible outcomes.

For each game option in the exercise, we calculate the probability using combinations. For instance, if we need exactly three red balls, the probability is \( \frac{\binom{5}{3} \cdot \binom{15}{n-3}}{\binom{20}{n}} \). Each probability accurately represents the chances of meeting specific requirements.

Understanding probability calculations provides a clearer picture of each game's feasibility.
  • Formula for probability: \( \frac{\text{Favorable outcomes}}{\text{Total outcomes}} \)
  • Essential for determining winning chances in games.
Game Theory
Game theory examines strategies for making decisions in competitive situations. This concept is crucial for maximizing long-term benefits when playing games involving probability.In scenarios like the amusement park game, game theory prompts players to analyze potential outcomes and choose the option with the highest expected value or winning probability. By evaluating each option for different \( n \) values, players can optimize their strategy for maximum gain.

Game theory enhances decision-making by considering the strategic component of probability games beyond mere chance.
  • Focuses on strategic decision-making in competitive settings.
  • Involves analyzing expected values and winning probabilities.

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Most popular questions from this chapter

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Diffcrcncc cquation for nandom numbcrs. Simple random number generators are based on simulating difference equations. Here is a typical set of two equations: $$ \begin{aligned} &x_{n}=\left(a x_{n-1}+c\right) \bmod m \\ &y_{n}=x_{n} / m \end{aligned} $$ for \(n=1,2, \ldots .\) A seed \(x_{0}\) must be given to start the sequence. The numbers \(y_{1}, y_{2}, \ldots\), represent the random numbers and \(x_{0}, x_{1}, \ldots\) are "help" numbers. Although \(y_{n}\) is completely deterministic from (8.14)(8.15), the sequence \(y_{n}\) appears random. The mathematical expression \(p\) mod \(q\) is coded as \(p \%\) q in Python. Use \(a=8121, c=28411\), and \(m=134456 .\) Solve the system \((8.14)-\) (8.15) in a function that generates and returns \(N\) random numbers. Make a histogram to examine the distribution of the numbers (the \(y_{n}\) numbers are randomly distributed if the histogram is approximately flat). Name of program file: diffeq_random.py.

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