Harry Shannon
As I walked with my wife to our local library, we passed under a black walnut tree. Eileen said she was concerned about being hit on the head by a falling walnut. It’s that time of year. The nuts fall in their shells that are very hard and can make quite a noise depending on what they land on. (If one fell on your head, though, more noise would come from you than from the collision itself.)
Naturally, I wanted to reassure Eileen, saying it was unlikely she would be hit. Then I realized I could try to estimate how unlikely. I had to make several assumptions, some of which would be pretty accurate, others less so. Here goes …
Suppose the tree sheds 1,000 nuts and that takes close to a month, so about 40 per day. The distance on the ground under the tree is around 10 m which takes about 7-8 seconds to walk. The chance that a nut will fall during that time works out to 1 in 288, let’s round that to 1 in 300.
Now we need to allow for the area of the tree and the area Eileen’s head takes up. The tree canopy radius is about 5 m, so its area is 25π m2. The radius of a head is about 0.1 m (yes, I thought that seemed too low, but it’s about right). The area is 0.01π m2, about 1 in 2,500 of the tree canopy. That’s the probability that at the instant when a nut falls, Eileen is under that part of the tree.
Multiplying these together, I estimate that on any occasion she walks under the tree, the probability that Eileen’s head is hit by a walnut shell is only 1 in 750,000.
Of course, she could be hit elsewhere on her body, which would roughly double the likelihood. It would also double if we walk home on the same route. And you could question my assumptions. Even so, I don’t think Eileen needs protective gear.