On average, 63 of the 100 detected electrons will have originated from within a distance of 1λ of the surface.

Rationale

As stated previously the probability of a detected electron originating from a particular depth, P'(d), is proportional to P(d), i.e.

    P'(d) = c. exp ( -d / λ )

where the value of c is determined by the normalisation condition that the electron must come from somewhere ! - i.e. that the integral of P'(d) over all possible values of d from zero to infinity must be equal to unity. This gives c = (1/λ ).

The probability of an electron originating from d < λ is given by integrating P'(d) over the range d = 0 to d = λ , i.e.

    ⇒   P'(d < λ) = -1 [ exp(-1) - exp(0) ] = ( 1 - exp(-1)) = 0.63

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