Question

The probability of at least three successes in six trials of a binomial experiment in which p = 1/2

Answer 1

http://openstudy.com/study#/updates/53127432e4b0966c639ba915
$$
\large{
P(X \ge3)=1-P(X<3)\\
=1-P(X\le2)\\
=1-(P(X=0)+P(X=1)+P(X=2))\\
=1-\sum_{i=0}^2 \binom{6}{i}p^iq^{6-i}\\
=1-\sum_{i=0}^2 \binom{6}{i}0.5^6\\
=1-0.5^6\sum_{i=0}^2 \binom{6}{i}\\
=1-0.5^6\left(\binom{6}{0}+\binom{6}{1}+\binom{6}{2}\right )\\
=1-0.5^6\left(1+6+15\right )\\
=1-0.5^6\left(32\right )\\
=1-0.5^6\left(2^5\right )\\
=1-\cfrac{2^5}{2^6}\\
=1-\cfrac{1}{2}\\
=\cfrac{1}{2}
}
$$