## A neat expected value problem from Expii

[sorry for the quick write up – I got asked to help out with my son’s archery class today, so I just decided to publish this one as it was when I get asked to help . . . ]

I saw a neat expected value problem from Expii yesterday. In case you’ve not see their site, here’s the link to their main site:

Expii’s front page

and here’s a direct link to the problem:

A neat expected value problem from Expii

The problem goes like this:

“You are planting some trees as environmental action for Earth Day. At each of 200 spots around a circle, you place a seed. Each seed will sprout into a small tree with probability 1/2. Sadly, some of these small trees will die. In particular, a small tree dies if it has another small tree as its neighbor, because they will be fighting for sunlight.

What is the expected value of the number of trees that are still alive at the end of the year?”

I thought this would be a great problem to discuss with the boys. We just got back from a vacation in San Diego and my younger son was still on west coast time, though, so I just talked through this one with my older son.

First I introduced the problem and we double checked that he understood it:

Next we discussed some simple cases to see if we could get our arms around the problem:

Now we moved on to the general case. My son understood some of the main ideas about the problem, but made a small mistake at the end that led to a very small expected value.

Finally, we wrapped up by looking at the error at the end of the last video and trying to calculate the expected value slightly more carefully: