I'm Optimistic With The Results!

Rough solution :

For a binomial distribution , the probability of obtaining exactly $k$ successes out of $n$ trials with success rate of each trial be $p$ is $P(k) = \dbinom nk p^k (1-p)^{n-k} \; .$

So we want to find $P(6) + P(7) + P(8)$ , which equals to

$\dfrac1{2^8} \left [ \dbinom 86 + \dbinom 87 + \dbinom 88 \right ] = \boxed{\dfrac{37}{256}} \; .$