How to Estimate Chance with Dice Rolls Using Convolutions and Recursion

In today's edition of Data Science Code in Python + R we're going to tackle a fascinating concept. We're diving into the world of convolutions and recursion, two powerful tools you can utilize in your data science toolkit. And we're going to do it with something fun: a dice-rolling exercise!?

To start off, let's break down these two terms. Convolution is a mathematical operation on two functions that produces a third function. It’s a way of combining two sets of information. For data scientists, this comes up often in the context of image and signal processing, but it has other interesting applications as well.

On the other hand, recursion is a method of solving problems by having functions call themselves. In other words, a recursive function solves a problem by solving smaller instances of the same problem.

Let's now talk about how we can apply these concepts to our dice rolling scenario.

Step-by-Step Convolution and Recursion:

Consider a single die roll of a six-sided die. The outcomes can range from 1 to 6, each with equal probability of 1/6. We can represent this as a probability vector (single_die_prob) where each element corresponds to the probability of the respective outcome. The probability of getting any particular sum when rolling a single die is straightforward - it's simply 1/6 for each of the possible outcomes (1 through 6).

# Generate the probability vector for a single die in R
single_die_prob = rep(1/num_sides, num_sides)        

But what happens when you roll more than one die and want to know the probability of getting a particular sum? This is where convolutions come into play!

The convolution of the distributions of two independent random variables gives us the distribution of their sum. So, if we roll two dice and sum the result, we can get the probability distribution of the sum by convolving the distributions of the individual rolls.?

# Generate the probability vector for the sum of two dice roll in R

two_dice_sum_prob = convolve(single_die_prob, single_die_prob, type = "open"))        

This gives us the probability distribution for the sum of two dice. The process can be repeated for more dice. For three dice, we take the convolution of the distribution for two dice with the distribution for one die:

# Generate the probability vector for the sum of three dice rolls in R

three_dice_sum_prob = convolve(two_dice_sum_prob, single_die_prob, type = "open"))        

If we have many dice, we don't want to manually write out each step like this. This is where recursion comes in!

Recursion allows us to automate this process by convolving the distribution for n dice with the distribution for one die to get the distribution for n + 1 dice. This is done using the `Reduce` function in R, which applies a function recursively over a list.?

# Generate the probability vector for the sum of multiple dice roll in R

dice_sum_prob = Reduce(function(x, y) convolve(x, y, type = "open"), rep(list(single_die_prob), num_dice))        

And that's it! We've calculated the probability distribution for the sum of dice rolls using convolutions and recursion.

Lastly, we can sum up the probabilities for sums greater than a target value:

# Calculate the probability of the sum being greater than target_sum in R

favorable_prob = sum(dice_sum_prob[(target_sum - num_dice + 2):length(dice_sum_prob)])        

Putting it Together with Python and R

The output for the R code is shown in the cover image for this article. We can use a similar approach in Python with a few modifications.

# Calling our Python function
estimate_dice_roll_chances(num_dice = 5, sum_greater_than = 20)}        

We see a similar result rendered with Seaborn and Matplotlib using our Python function for the case where we have 5 standard six-sided dice and want to know the chances a roll will sum to greater than 20.

Other Applications

The beauty of this approach is that it isn't limited to dice rolls. Anytime you're dealing with independent random variables and you're interested in the probability distribution of their sum, convolution can come into play. This can be applied in a wide variety of scenarios:

1. Signal Processing: In the field of digital signal processing, convolutions are used to apply filters to signals.
2. Image Processing: Convolution is at the heart of convolutional neural networks (CNNs), a class of deep learning models most commonly applied to analyzing visual imagery. They're used in feature extraction which is vital for image recognition tasks.
3. Natural Language Processing (NLP): Convolutional neural networks can also be used in text analysis to capture the sequence of words in a sentence or phrases.
4. Time series analysis: The moving average of a time series can be obtained through the convolution of the time series data and a sequence of weights.

Wrapping Up

So, there you have it, a step-by-step walkthrough of convolution and recursion, set in the context of a dice rolling problem. I hope this helped you grasp these fundamental concepts. With these tools in your data science toolkit, you can now approach many problems in a new and more efficient way.

Remember, as with anything in data science, practice is key. Try experimenting with these concepts in different scenarios and exploring further applications.

Happy coding and remember to like and leave a comment if you found this article interesting or helpful!

~ Matt