Often in physics we won't be able to predict motion of a bunch of objects exactly and so we resort to statistical methods. An example of a statistical approach is the resulting profile of a bunch of disks dropped through a series of pegs. As each disk falls it encounters pegs and bounces to the left or the right of the peg with equal probability.

If we dropped many disks from the top through the pegs, what is the ratio of the number of disks in bin 6 to the number of disks in bin 4?

Wouldnt it matter how many bins there were, and the locations of those bins? I suspect the normal distribution is involved here, or the Poission distribution.

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To determine the ratio of the number of disks in bin 6 to the number of disks in bin 4, we need to understand the probabilities associated with the disk motion.

Each time a disk encounters a peg, it has an equal probability of bouncing to the left or the right. Since each peg has two possible outcomes, this represents a binary outcome.

When considering the number of disks in each bin, we can use a combinatorial approach to analyze the possible outcomes.

Let's consider a single disk and see how it can end up in bin 4 or bin 6:

1. For a disk to end up in bin 4, it needs to bounce to the right once after falling through bin 5 (so that it does not get redirected to bin 6). The probability of this occurring is 1/2 * 1/2 = 1/4.

2. For a disk to end up in bin 6, it needs to bounce to the left once after falling through bin 5. The probability of this occurring is also 1/2 * 1/2 = 1/4.

Therefore, the ratio of the number of disks in bin 6 to the number of disks in bin 4 would be determined solely by the number of disks initially dropped into the system, assuming all disks have an equal chance of following the specified probability distribution.

If we denote the number of disks dropped as N, then the ratio of the number of disks in bin 6 to bin 4 would simply be N * (1/4) / N * (1/4), which simplifies to 1/1 or 1:1.

Thus, the ratio of the number of disks in bin 6 to the number of disks in bin 4 would be 1:1, assuming enough disks are dropped that the effect of individual disks bouncing left or right equalizes over time.

To determine the ratio of the number of disks in bin 6 to the number of disks in bin 4, we can utilize a statistical approach.

First, let's consider the possible outcomes for each disk as it encounters the pegs. Since each disk has an equal probability of bouncing to the left or right of a peg, we can model the process as a binomial distribution. Each peg can be thought of as a binary choice, with equal chances for the disk to bounce left or right.

Now, let's analyze the disks' paths and how they relate to the bins. Since the disks are dropped from the top, they go through the pegs one by one, and each disk takes a unique path. As they fall, the disks will encounter different numbers of pegs, which will determine the final bin they end up in.

To determine the ratio of disks in bin 6 to bin 4, we need to assess the probabilities associated with each bin. Let's break it down step by step:

1. Calculate the total number of possible paths for a disk when encountering the pegs. Since each peg presents an independent binary choice (left or right), the number of possible paths is given by 2 raised to the power of the number of pegs.
For example, if there are 3 pegs, there would be 2^3 = 8 possible paths.

2. Determine the number of paths that result in a disk ending up in bin 6 or bin 4. To do this, we need to count how many unique paths result in the disk reaching each bin.
For bin 4, count the number of paths that pass through exactly 2 pegs (as bin 4 is the fourth bin, counting starts from the top).
For bin 6, count the number of paths that pass through exactly 4 pegs (as bin 6 is the sixth bin, counting starts from the top).

3. Calculate the ratio of disks in bin 6 to bin 4 by dividing the total number of paths leading to bin 6 by the total number of paths leading to bin 4.
Ratio = (Number of paths to bin 6) / (Number of paths to bin 4).

Since the exact number of pegs and bins is not specified in the question, you may need to provide that information to calculate the ratio accurately.