I recently stumbled upon a fascinating question on Reddit that caught my attention. The question was simple, yet it required some statistical thinking to arrive at the answer. Here’s the question: if you have 100 balls and choose 50 without replacement, then replace those 50 balls and repeat the process, how many unique balls will you have chosen on average?
At first glance, it seems like a straightforward problem, but the answer might surprise you. To understand the solution, let’s break it down step by step.
When you choose 50 balls from 100, you’re essentially sampling from a population of 100. Since you’re replacing the balls after the first round, the probability of choosing each ball remains the same for the second round. Now, here’s the crucial part: the number of unique balls you’ll choose in the second round is not 50, but slightly less than that.
To calculate the expected number of unique balls, we can use the concept of expected value. The expected value of a random variable is the long-run average value of that variable over many trials. In this case, the random variable is the number of unique balls chosen in each round. Using some statistical wizardry, we can calculate the expected value to be around 75. This means that, on average, you’ll choose around 75 unique balls after two rounds.
But what does this have to do with DNA? The original poster was trying to understand how much DNA from one parent would show up when two of their children take DNA tests. The connection lies in the fact that DNA is essentially a sampling process. When you inherit DNA from your parents, it’s like randomly sampling from their genetic material. By understanding the statistical principles behind this process, we can gain insights into how DNA is passed down through generations.
In conclusion, the answer to the original question is around 75 unique balls, which might seem counterintuitive at first. However, by applying statistical principles, we can uncover the underlying patterns and gain a deeper understanding of complex phenomena like DNA inheritance.