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. 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

To compute the expected number of trees that will die in one summer, we need to consider the number of trees in different positions along the line. The expected number of trees that will die in one summer is (n+1)/4.

Step-by-step explanation:

To compute the expected number of trees that will die in one summer, we need to consider the number of trees in different positions along the line. Let's assume there are n trees in total.

For the trees at the very front and back, they will die if they are shorter than their (only) neighbor. Therefore, there is a 50% chance that each of them will die.

For the rest of the trees, if a tree is shorter than both the trees directly in front and behind it, it does not receive enough sunlight to survive summer. The probability of this happening is (1/2)^2 = 1/4 for each tree. Since there are (n-2) such trees, the expected number of trees that will die in this case is (n-2)*(1/4) = (n-2)/4.

Putting it all together, the expected number of trees that will die in one summer is (1/2) + (1/2) + (n-2)/4 = (n+1)/4.