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Chapter 8: Problem 8

Decide if a dice game is fair. Somebody suggests the following game. You pay 1 unit of money and are allowed to throw four dice. If the sum of the eyes on the dice is less than 9 , you win 10 units of money, otherwise you lose your investment. Should you play this game? Answer the question by making a program that simulates the game. Name of program file: sum9_4dice.py.

Short Answer

Expert verified
Simulate the game to get the probability of winning and calculate the expected value; if it's negative, the game is not fair.

Step by step solution

01

Understand the Game Rules

You pay 1 unit to play the game, and if the sum of four dice rolls is less than 9, you win 10 units. Otherwise, you lose your initial 1 unit.
02

Calculate Probability of Winning

To determine the fairness of the game, first calculate the probability of the total sum of four dice being less than 9. The possible sums from rolling four dice range from 4 (all ones) to 24 (all sixes). We need to find the outcomes where the sum is less than 9.
03

Calculate Expected Value

The expected value (EV) tells us the average outcome of many trials. Calculate EV as: \[\text{EV} = P(\text{win}) \times \text{win amount} - P(\text{lose}) \times \text{lose amount}\]where \(P(\text{win})\) is the probability of winning (sum < 9), and \(P(\text{lose})\) is the probability of losing (sum \(\geq\) 9).
04

Simulate the Game

To find the probabilities, simulate a large number of dice throws using a Python program (`sum9_4dice.py`). Use random number generation to roll four dice repeatedly and keep track of how often the total sum is less than 9. Increase the number of simulations to get a more accurate probability.
05

Analyze Results and Conclude

Once you have the simulation results, calculate the probability of winning. Use this to find the expected value. If the expected value is positive, the game is fair or favorable; if negative, the game is not fair.

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Key Concepts

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

Expected Value
Expected value is a vital concept in probability that helps to assess the fairness or viability of a game. It represents the long-term average outcome you would expect if you were to play the game many times. In mathematical terms, the expected value (EV) is calculated by summing the products of each possible outcome's probability and its respective payoff or loss.

In the context of the dice game:
  • The winning scenario where the dice sum is less than 9 lets you earn 10 units, but since you paid 1 unit to play, your net gain is 9 units.
  • The losing scenario occurs when the dice sum is 9 or higher, which means you lose your initial 1 unit investment.
The formula to calculate the expected value in this scenario is:
\[\text{EV} = P(\text{win}) \times 9 - P(\text{lose}) \times 1\]

By determining these probabilities, anyone can use the expected value to decide whether or not they should engage in the game.
Simulation
Simulating a game can be a powerful approach to understanding the chances of winning or losing when mathematics gets complex or when you need a hands-on experience. In this exercise, simulation involves mimicking the dice game by replicating the process of throwing the dice multiple times and observing the results, which might be difficult to calculate purely analytically for each potential roll.

This approach has several steps:
  • Define the conditions of the game - in this case, rolling four dice and checking sums.
  • Use a tool (like Python) to simulate many dice throws 鈥� ideally thousands of times to get a reliable probability estimate.
  • Count how many times the sum is less than 9 (a win) versus greater or equal to 9 (a loss).
We use these results to estimate the probabilities needed to determine if the game is fair. By focusing on repetitive outcomes, simulations help reveal patterns and probabilities that inform the expected value.
Dice Games
Dice games are a classic form of gambling and random chance games, using dice to produce an outcome based on probability. The attractiveness lies in their simplicity and the randomness provided by the dice, making each game unpredictable yet exciting.

In this particular exercise, the game rules require rolling four six-sided dice with the objective to have a sum less than 9 to win. Let鈥檚 break down the dynamics:
  • The lowest sum of four dice rolls is 4 (all ones), and the highest is 24 (all sixes).
  • Calculations or simulations need to determine how often results meet the winning condition.
The quick calculation of odds and outcomes in dice games helps players understand risk and make informed decisions about playability. Learning to evaluate dice games with probability provides players an edge or鈥攁t least鈥攔easonable expectations.
Python Programming
Python programming is integral in simulating probability events, such as our dice game, thanks to its simplicity and the powerful library functions it offers. Python's ease of use for beginners makes it ideal for educational purposes and practical applications in probability analysis.

In this scenario, the Python program `sum9_4dice.py` performs the following tasks:
  • Use the `random` library to simulate rolling four six-sided dice. The `randint()` function generates a random integer between 1 and 6 to simulate each die.
  • Repeat the simulation for a large number of trials to estimate probabilities accurately.
  • Count the outcomes where the sum is less than 9 and compare it with other possible sums to calculate the probability of winning.
Finally, Python can help to calculate and interpret the expected value based on the results. By doing so, it makes an abstract probability concept tangible and understandable, especially when exploring fairness in games.

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Most popular questions from this chapter

Estimate the probability in a dice game. Make a program for estimating the probability of getting at least one 6 when throwing \(n\) dice. Read \(n\) and the number of experiments from the command line. (To verify the program, you can compare the estimated probability with the exact result \(11 / 36\) when \(n=2\).) Name of program file: one6_ndice.py.

Compute velocity and acceleration. In a laboratory experiment waves are generated through the impact of a model slide into a wave tank. (The intention of the experiment is to model a future tsunami event in a fjord, generated by loose rocks that fall into the fjord.) At a certain location, the elevation of the surface, denoted by \(\eta\), is measured at discrete points in time using an ultra-sound wave gauge. The result is a time series of vertical positions of the water surface elevations in meter: \(\eta\left(t_{0}\right), \eta\left(t_{1}\right), \eta\left(t_{2}\right), \ldots, \eta\left(t_{n}\right)\). There are 300 observations per second, meaning that the time difference between to neighboring measurement values \(\eta\left(t_{i}\right)\) and \(\eta\left(t_{i+1}\right)\) is \(h=1 / 300\) second. Write a Python program that accomplishes the following tasks: 1\. Read \(h\) from the command line. 2\. Read the \(\eta\) values in the file src/random/gauge. dat into an array eta. 3\. Plot eta versus the time values. 4\. Compute the velocity \(v\) of the surface by the formula $$ v_{i} \approx \frac{\eta_{i+1}-\eta_{i-1}}{2 h}, \quad i=1, \ldots, n-1 $$ Plot \(v\) versus time values in a separate plot. 5\. Compute the acceleration \(a\) of the surface by the formula $$ a_{i} \approx \frac{\eta_{i+1}-2 \eta_{i}+\eta_{i-1}}{h^{2}}, \quad i=1, \ldots, n-1 $$ Plot \(a\) versus the time values in a separate plot.

Compute \(\pi\) by a random sum. Let \(x_{0}, \ldots, x_{N}\) be \(N+1\) uniformly distributed random numbers between 0 and \(1 .\) Explain why the random sum \(S_{N}=\sum_{i=0}^{N} 2\left(1-x_{i}^{2}\right)^{-1}\) is an approximation to \(\pi\). (Hint: Interpret the sum as Monte Carlo integration and compute the corresponding integral exactly by hand.) Make a program for plotting \(S_{N}\) versus \(N\) for \(N=10^{k}, k=\) \(0,1 / 2,1,3 / 2,2,5 / 2, \ldots, 6\). Write out the difference between \(S_{10^{6}}\) and pi from the math module. Name of program file: MC_pi_plot.py.

Choose random colors. Suppose we have eight different colors. Make a program that chooses one of these colors at random and writes out the color. Hint: Use a list of color names and use the choice function in the random module to pick a list element. Name of program file: choose_color.py.

Find the expected waiting time in traffic lights. A driver must pass 10 traffic lights on a certain route. Each light has a period red-yellow-green-yellow of two minutes, of which the green and yellow lights last for 70 seconds. Suppose the driver arrives at a traffic light at some uniformly distributed random point of time during the period of two minutes. Compute the corresponding waiting time. Repeat this for 10 traffic lights. Run a large number of routes (i.e., repetitions of passing 10 traffic lights) and let the program write out the average waiting time. Does the computed time coincide with what you would expect? Name of program file: waiting_time.py.
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