Question

The probability that a person is not a sportsperson is $1/6$. Then the probability that out of 6 members of the family, 5 are sportspersons is ______.

• A. $(5/6)^5$
• B. $6(5/6)^5$
• C. $5(5/6)^6$
• D. $(5/6)^6$

Hint:

Be careful! It is extremely tempting to just pick $6(5/6)^5$ because of the ${}^{n}C_r$ factor. Always fully write out the $q^{n-r}$ failure term, as it often cancels out coefficients like it did here!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This follows a Binomial Distribution $B(n, p)$. $n = 6$. Probability of "Not Sportsperson" $q = 1/6$. Probability of "Sportsperson" $p = 1 - 1/6 = 5/6$.

Step 2: Formula Application:
$P(X = 5) = \binom{n}{r} p^r q^{n-r}$. $P(X = 5) = \binom{6}{5} (5/6)^5 (1/6)^1$.

Step 3: Explanation:
$\binom{6}{5} = 6$. $P(X = 5) = 6 \times \frac{5^5}{6^5} \times \frac{1}{6} = \frac{6 \times 5^5}{6^6} = \frac{5^5}{6^5} = (5/6)^5$.

Step 4: Final Answer:
The probability is $(5/6)^5$.