Question

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

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

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.

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.

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).

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.

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.

Key concepts

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 — 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’s 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—at least—reasonable 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.