Evaluate the probability of finding an electron in a small region of a hydrogen 1s orbital

Use the equation for the wavefunction for a 1 s H atom (page 25 of the 5th edition of the textbook, page 21 of the 4th edition of textbook) and square it: Ψ2r,Θ,Φ( ) =1πa03e−2ra0to get the probability density.

Then take the ratio of the probability density at r=0.55 ao and at r=0.Ψ 20.55ao,Θ, Φ( )Ψ20,Θ,Φ( )=1πao3e−2 0.55ao( )ao1πao3e−2 0( )ao=e−2 0.55ao( )aoe−2 0( )ao==e−2(0.55)1=0.3329 = 0.33

The ratio of the probability density at r = 0.55 a0 to that at r = 0 is 0.33.

See attachment for better understanding.