The probability that a person is not a sportsperson is $\frac{1}{6}$. Then the probability that out of the 6 members of the family, 5 are sportspersons is

The probability that a person is not a sportsperson is given as $P(\text{not } S) = \frac{1}{6}$. The complement yields the probability of being a sportsperson: $P(S) = 1 - \frac{1}{6} = \frac{5}{6}$.

Finding the probability that exactly 5 out of 6 family members are sportspersons is a binomial probability problem. Using the binomial formula $P(X = k) = C(n, k) p^k q^{n-k}$ with $n = 6$, $k = 5$, $p = \frac{5}{6}$, and $q = \frac{1}{6}$, we compute:

$P(X = 5) = C(6, 5) \left(\frac{5}{6}\right)^5 \left(\frac{1}{6}\right)^1$

Since $C(6, 5) = 6$, this simplifies to:

$P(X = 5) = 6 \cdot \frac{5^5}{6^5} \cdot \frac{1}{6} = \frac{5^5}{6^5} = \left(\frac{5}{6}\right)^5$

This matches option A.