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Question
A fair coin is tossed 8 times, find the probability of at most six heads.
Solution
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 \[q = 1 - \frac{1}{2} = \frac{1}{2}\]
Probability of getting at most 6 heads
\[= P\left( X \leq 6 \right)\]
\[ = 1 - \left[ P\left( X = 7 \right) + P\left( X = 8 \right) \right]\]
\[ = 1 - \left[ {}^8 C_7 \left( \frac{1}{2} \right)^8 +^8 C_8 \left( \frac{1}{2} \right)^8 \right]\]
\[ = 1 - \left( \frac{8}{256} + \frac{1}{256} \right)\]
\[ = 1 - \frac{9}{256}\]
\[ = \frac{247}{256}\]
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