Hello!
The probability of being between some c and d is %7D%5Cright%7C%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D. "int_c^d |Psi(x)|^2 dx.")
Therefore the probability in question is
%7D%5Cright%7C%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%7B%5Cint_%7B%7B%7Ba%7D-%7B0.027%7D%7Ba%7D%7D%7D%5E%7B%7B%7Ba%7D%2B%7B0.027%7D%7Ba%7D%7D%7D%7D%5Cfrac%7B%7B4%7D%7D%7B%7B%7Ba%7D%7D%5E%7B%7B3%7D%7D%7D%7B%7Bx%7D%7D%5E%7B%7B2%7D%7D%7B%7Be%7D%7D%5E%7B%7B-%5Cfrac%7B%7B%7B2%7D%7Bx%7D%7D%7D%7B%7Ba%7D%7D%7D%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D "int_(a-0.027a)^(a+0.027a) |Psi(x)|^2 dx =int_(a-0.027a)^(a+0.027a) 4/a^3 x^2 e^(-(2x)/a) dx")
(
must be positive, so bounds of integration are also positive).
To compute the indefinite integral of 
we can use integration by parts twice: differentiate 
and then 
and integrate the exponent. Let's perform this:
Hello!
The probability of being between some c and d is Therefore the probability in question is
( must be positive, so bounds of integration are also positive).
To compute the indefinite integral of we can use integration by parts twice: differentiate and then and integrate the exponent. Let's perform this:
%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D. "= -a/2 x^2e^(-(2x)/a) + a/2 int (2xe^(-(2x)/a)) dx.")
Then 
and the remaining integral is equal to
So the total indefinite integral is equal to %7D%2B%7BC%7D%2C "-a/4 e^(-(2x)/a)(a^2+2ax+2x^2)+C,")
and the probability is
%7D%7D%7D%7B%5Cleft(%7B%7Ba%7D%7D%5E%7B%7B2%7D%7D%2B%7B2%7D%7B%7Ba%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7B1%7D%2B%7B0.027%7D%5Cright)%7D%2B%7B2%7D%7B%7Ba%7D%7D%5E%7B%7B2%7D%7D%7B%7B%5Cleft(%7B1%7D%2B%7B0.027%7D%5Cright)%7D%7D%5E%7B%7B2%7D%7D%5Cright)%7D-%5Cright.%7D "-4/a^3*a/4 (e^(-2(1+0.027))(a^2+2a^2(1+0.027)+2a^2(1+0.027)^2)-")
%7D%7D%7D%7B%5Cleft(%7B%7Ba%7D%7D%5E%7B%7B2%7D%7D%2B%7B2%7D%7B%7Ba%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7B1%7D-%7B0.027%7D%5Cright)%7D%2B%7B2%7D%7B%7Ba%7D%7D%5E%7B%7B2%7D%7D%7B%7B%5Cleft(%7B1%7D-%7B0.027%7D%5Cright)%7D%7D%5E%7B%7B2%7D%7D%5Cright)%7D%7B%5C)%7D%7B%5C)%7D. "- e^(-2(1-0.027))(a^2+2a^2(1-0.027)+2a^2(1-0.027)^2))).")
vanishes and remains
%7D%7D%7D%7B%5Cleft(%7B1%7D%2B%7B2%7D%7B%5Cleft(%7B1%7D-%7B0.027%7D%5Cright)%7D%2B%7B2%7D%7B%7B%5Cleft(%7B1%7D-%7B0.027%7D%5Cright)%7D%7D%5E%7B%7B2%7D%7D%5Cright)%7D- "e^(-2(1-0.027))(1+2(1-0.027)+2(1-0.027)^2) -")
%7D%7D%7D%7B%5Cleft(%7B1%7D%2B%7B2%7D%7B%5Cleft(%7B1%7D%2B%7B0.027%7D%5Cright)%7D%2B%7B2%7D%7B%7B%5Cleft(%7B1%7D%2B%7B0.027%7D%5Cright)%7D%7D%5E%7B%7B2%7D%7D%5Cright)%7D%5Capprox%7B0.0292%7D. "- e^(-2(1+0.027))(1+2(1+0.027)+2(1+0.027)^2) approx 0.0292.")
This is the answer.
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