Let A be a set of all 4 -digit natural numbers whose exactly one...

$n\left(s\right) = n$( when $7$ appears on thousands place ) $+ n$($7$ does not appear on thousands place)

$= 9 \times 9 \times 9 + 8 \times 9 \times 9 \times 3$

$= 33 \times 9 \times 9$

$n\left(E\right) = n$( last digit $7 \text{ and } 7$ appears once)$+ n$( last digit $2$ when $7$ appears once)

$= 8 \times 9 \times 9 + \left(9 \times 9 + 8 \times 9 \times 2\right)$

$∴ P\left(E\right) = \dfrac{8 \times 9 \times 9 + 9 \times 25}{33 \times 9 \times 9} = \dfrac{97}{297}$