Question

Throw dice and compute a small probability. Compute the probability of getting 6 eyes on all dice when rolling 7 dice. Since you need a large number of experiments in this case (see the first paragraph of Chapter 8.3), you can save quite some simulation time by using a vectorized implementation. Name of program file: roll_7dice.py.

Short answer

The probability is \( \frac{1}{279936} \). It is a rare event.

Step-by-step solution

Understanding the Problem

We need to find the probability of rolling exactly seven sixes when rolling seven dice. Each die has six faces, numbered 1 through 6.

Calculate the Probability of a Single Die Roll

For a single die to show a six, the probability is \( \frac{1}{6} \). Only one face out of the six possibilities is a six.

Calculate Probability for All Dice to Show Six

To find the probability of all seven dice showing a six simultaneously, we multiply the probability of a single die showing a six by itself seven times: \( \left( \frac{1}{6} \right)^7 \).

Perform the Calculation

Perform the actual computation to find \( \left( \frac{1}{6} \right)^7 = \frac{1}{6^7} \). Compute \( 6^7 \) to find 279936, hence the probability is \( \frac{1}{279936} \).

Interpret the Result

The probability signifies how likely it is to roll all sixes with seven dice simultaneously, which is a very rare event given the high denominator.

Key concepts

Dice Rolling

Dice rolling involves shaking a die and letting it rest to reveal a random number on one of its faces. Each face of a die is equally likely to appear. A standard die is a cube with faces numbered from 1 to 6.
In probability problems involving dice, you often need to calculate the likelihood of certain outcomes. For instance, in our example, we want to know the probability that every die in a set of seven dice will land on the number 6.
Each face showing a particular number, like a 6, has a probability of \( \frac{1}{6} \). This is because there is one favorable outcome (rolling a 6) out of six possible outcomes (the numbers 1 through 6).
When rolling multiple dice, if you want to calculate the probability of each die showing a specific number, you need to multiply the probability of the single event by itself, once for each die. This is because each roll is an independent event, meaning the outcome of one roll doesn't affect the other.

Vectorized Implementation

A vectorized implementation in computing is an efficient way to perform operations on data arrays using a single operation instead of looping through individual elements. This approach utilizes the power of array-based processing, often found in languages like Python with libraries such as NumPy.
The main advantage of vectorized implementation is a significant reduction in execution time, which is crucial for simulation tasks that involve large datasets or numerous calculations. By applying operations to entire arrays at once, programs can run faster because they avoid repetitive "for" loops, working with entire collections of data instead.
In the context of rolling dice, using a vectorized implementation can create thousands or millions of dice roll outcomes almost instantly. For example, instead of simulating each roll of the dice in a loop, you can simulate the roll of seven dice several thousand times simultaneously, significantly speeding up computation.

Simulation

Simulation is a technique used to mimic the operation of real-world processes or systems. In probability and statistics, simulations are often used to approximate answers to questions that are impractical to calculate exactly using analytical methods.
By rolling dice a large number of times and recording the outcomes, we can estimate probabilities. This way, we can test and verify theoretical probabilities by seeing how often certain outcomes occur.
Setting up a simulation involves setting parameters for your `experiment`, such as the type and number of dice and the number of simulations to run. Through simulation, you can quickly test the probability of events like rolling seven sixes with seven dice, allowing you to understand complex probability distributions in a more intuitive way.

Mathematical Computation

Mathematical computation involves performing operations to solve mathematical problems, often using algorithms or numerical techniques to efficiently calculate results. In probability problems like dice rolling, computation plays a critical role in deriving probabilities and validating results.
For example, to calculate the probability of rolling all sixes on seven dice, you compute the probability of a single die roll showing a six and then raise it to the power of seven: \( \left( \frac{1}{6} \right)^7 \). This calculation shows how rare the event is, as reflected in the small probability value of \( \frac{1}{279936} \).
Computational tools and environments like Python can simplify these calculations through coding, which allows for fast computations and quick what-if analyses. These tools make it easier to carry out intricate operations without manually performing each step, enabling learners to focus on understanding the underlying concepts.