# Probability Experiment with Caffeinated Sloths

## At the end of the experiment, what is the expected number of trees which have at least two sloths?

Based on Ran Daman's **hypothesis** the expected number of trees with at least two sloths in the experiment is one.

In Ran Daman's experiment, each **sloth** selects a tree uniformly at random, according to his hypothesis. The expected number of trees with at least two sloths can be determined by calculating the probability of two sloths choosing the same **tree** and summing this probability for each tree.

Since each sloth chooses a tree independently and randomly, the **probability** of two sloths selecting the same tree is 1 divided by the total number of trees, denoted as m. Therefore, the probability is 1/m.

Considering this, the expected number of trees with at **least** two sloths is obtained by multiplying the total number of trees (m) by the probability of two sloths selecting the same tree, which is 1/m. Simplifying this expression gives an expected value of 1.

Hence, based on **Ran's** hypothesis, we can expect that at the end of the experiment, there will be one tree with at least two sloths.

When Ran Daman performed the experiment with caffeinated sloths, his hypothesis suggested that each sloth would select a tree randomly and climb that tree. Therefore, the expected number of trees with at least two sloths was calculated based on the probability of two sloths choosing the same tree.

By considering the randomness of sloths' tree selection and the total number of trees in the experiment, it was determined that the expected value for trees with at least two sloths is one. This shows the distribution and behavior of sloths in the experiment as predicted by Ran's hypothesis.