Question

A fair coin is tossed 8 times, find the probability of at most six 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 at most 6 heads

\[= P\left( X \leq 6 \right)\]

\[ = 1 - \left[ P\left( X = 7 \right) + P\left( X = 8 \right) \right]\]

\[ = 1 - \left[ {}^8 C_7 \left( \frac{1}{2} \right)^8 +^8 C_8 \left( \frac{1}{2} \right)^8 \right]\]

\[ = 1 - \left( \frac{8}{256} + \frac{1}{256} \right)\]

\[ = 1 - \frac{9}{256}\]

\[ = \frac{247}{256}\]