Question

A group of trees, all with distinct heights, have been planted uniformly randomly in a line. However, due to overcrowding, because some trees are taller than others, certain trees will struggle to receive sunlight. We will determine how many of the trees are expected to die due to this lack of sunlight. (a) If any tree is shorter than both the trees directly in front and behind of it, then it does not receive enough sunlight to survive summer. The trees at the very front and back will die if they are shorter than their (only) neighbor. At the beginning of summer, there are n trees. Compute the expected number of trees that will die in one summer.

Answer 1

The expected number of trees that will die in one summer, based on the conditions specified, is
`\((n+1)/(4)\),` where
`\(n\)` represents the initial number of trees planted uniformly in a line.

To compute the expected number of trees that will die in one summer, let's analyze the probabilities of a tree dying based on its position within the line of trees.

Consider the scenario:

Trees are planted uniformly randomly in a line.

If any tree is shorter than both the trees directly in front and behind it, it will not receive enough sunlight and will die.

The trees at the very front and back will die if they are shorter than their only neighbor.

Let
`\(X\)` be the random variable representing the number of trees that die in one summer out of
`\(n\)` trees initially planted.

For each tree in the line (except the first and last), the probability of a tree dying is
`\(P(\text{die}) = (1)/(4)\)` , as it needs to be shorter than both its neighbors (front and back) for it to die.

For the first and last trees:

For the first tree, the probability of dying is
`\(P(\text{die})` =
`(1)/(2)\)` as it has only one neighbor.

For the last tree, the probability of dying is
`\(P(\text{die})` =
`(1)/(2)\)` as it has only one neighbor.

The expected number of trees that will die
`(\(E(X)\))` can be calculated as the sum of individual probabilities of a tree dying:

`\[ E(X) = P(\text{tree 1 dies}) + \sum_(i=2)^(n-1) P(\text{tree i dies}) + P(\text{tree n dies}) \]`

`\[ E(X) = (1)/(2) + (n-2) * (1)/(4) + (1)/(2) \]`

`\[ E(X) = (n+1)/(4) \]`

Hence, the expected number of trees that will die in one summer is
`\((n+1)/(4)\)` based on the given conditions.