Solving the 3Blue1Brown String Probability Problem (Without AI)

Let’s practice data science thinking through a probability problem. The 3Blue1Brown YouTube channel, created by Grant Sanderson, recently posted a video on a probability problem that has garnered significant attention. The problem involves a string of blue and brown beads, and we’re tasked with finding the probability of a specific arrangement occurring. In this article, we’ll explore the problem and provide a solution using basic probability concepts and data science thinking.

The Problem

Imagine a string of 4 beads, with 3 blue beads and 1 brown bead. The beads are arranged randomly, and we want to find the probability that the brown bead is surrounded by blue beads on both sides. In other words, we’re looking for the probability of the arrangement “blue-brown-blue” or “blue-blue-brown-blue” within the string.

Breaking Down the Problem

To tackle this problem, let’s break it down into smaller, more manageable parts. We can start by considering the total number of possible arrangements of the 4 beads. Since there are 3 blue beads and 1 brown bead, the total number of arrangements can be calculated using the formula for permutations of a multiset:

n! / (n1! \* n2! \* … \* nk!)

where n is the total number of objects (beads), and n1, n2, …, nk are the numbers of each type of object (blue and brown beads). In this case, we have:

4! / (3! \* 1!) = 4

So, there are 4 possible arrangements of the 4 beads:

• Blue-Blue-Blue-Brown
• Blue-Blue-Brown-Blue
• Blue-Brown-Blue-Blue
• Brown-Blue-Blue-Blue

Identifying the Favorable Outcomes

Now, let’s identify the arrangements that satisfy the condition of the brown bead being surrounded by blue beads on both sides. From the list above, we can see that there are 2 favorable outcomes:

• Blue-Blue-Brown-Blue
• Blue-Brown-Blue-Blue

Calculating the Probability

With the total number of possible arrangements and the number of favorable outcomes, we can now calculate the probability. The probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes

In this case, the probability is:

Probability = 2 / 4 = 1/2

So, the probability that the brown bead is surrounded by blue beads on both sides is 1/2 or 50%.

Conclusion

In this article, we’ve applied basic probability concepts and data science thinking to solve the 3Blue1Brown string probability problem. By breaking down the problem, identifying the favorable outcomes, and calculating the probability, we’ve arrived at a solution without relying on artificial intelligence. This problem highlights the importance of critical thinking and analytical skills in data science, and we hope this solution inspires you to practice and develop your own problem-solving abilities.

The post Solving the 3Blue1Brown String Probability Problem (Without AI) appeared first on Towards Data Science.