If a coin is tossed 8 times and getting a head is considered as succe…

The correct answer is Option (3) → (A), (B), and (C) only

The number of heads X in 8 tosses of a fair coin follows the binomial distribution with \( n=8 \) and \( p=q=\frac{1}{2} \):

    \( \displaystyle P(X=r)={}^{8}C_{r}\left(\frac{1}{2}\right)^{8} \)

    \( \displaystyle =\frac{{}^{8}C_{r}}{256} \)

    \( \displaystyle P(X=4)=\frac{70}{256}=\frac{35}{128} \)

    \( \displaystyle P(X=6)=\frac{28}{256}=\frac{7}{64} \)

    \( \displaystyle \text{Mean}=np=8\times\frac{1}{2}=4 \)

    \( \displaystyle \text{Variance}=npq=8\times\frac{1}{2}\times\frac{1}{2}=2 \)

So P(4 heads) = \( \frac{35}{128} \), the mean is 4, the variance is 2 and P(6 heads) = \( \frac{7}{64} \).

Checking the statements

(A) The probability of getting 4 heads is \( \frac{35}{128} \). — Correct. \( {}^{8}C_{4}=70 \) and \( \frac{70}{256}=\frac{35}{128} \).

(B) The mean of the number of successes is 4. — Correct. \( np=8\times\frac{1}{2}=4 \).

(C) The variance of the number of successes is 2. — Correct. \( npq=8\times\frac{1}{4}=2 \).

(D) The probability of getting 6 heads is \( \frac{9}{64} \). — Incorrect. It is \( \frac{28}{256}=\frac{7}{64} \), not \( \frac{9}{64} \).