The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through...

Hello!

First we determine the probability `p` of the event that one electron will land in a specified strip. Then the expected number of such electrons will be `p*N,` where `N` is the total number of electrons.

By the definition of a probability density `P(x),` the probability of being between `a` and `b` is equal to  `int_a^b P(x) dx.` Here `a=-0.1` mm and `b=0.1` mm. Because the given density is an even function,  `p = 2 int_0^0.1 P(x) dx.`

It is simple to write a formula for `P(x)` for `x` between 0 and 3, it is `P(x)=1/3 - x/9` (a straight line). Therefore

`p = 2 int_0^0.1 (1/3 - x/9) dx = 2 (x/3 - x^2/18)|_(x=0)^0.1 = 2*(0.1/3-0.01/18) approx 0.0656.`

And the expected number is about  `4100*0.0328 approx 269.` This number is dimensionless.

Note that this number is the most probable, but the neighboring numbers are also very probable.