Question

A fair coin is tossed 8 times, find the probability of exactly 5 heads  .

Answer 1

Let X denote the number of heads obtained when a fair is tossed 8 times.
Now, X is a binomial distribution with n = 8,\[p = \frac{1}{2}\] and \[q = 1 - \frac{1}{2} = \frac{1}{2}\]

\[\therefore P\left( X = r \right) =^8 C_r \left( \frac{1}{2} \right)^{8 - r} \left( \frac{1}{2} \right)^r =^8 C_r \left( \frac{1}{2} \right)^8 , r = 0, 1, 2, . . . , 8\]

Probability of getting exactly 5 heads = \[P\left( X = 5 \right) =^8 C_5 \left( \frac{1}{2} \right)^8 = \frac{56}{256} = \frac{7}{32}\]