Please solve and explain THanks Suppose a student answers 20

Please solve and explain! THanks

Suppose a student answers 20 true/false questions completely at random. All questions are independent from each other. Use a normal approximation to estimate the probability of getting no more than 8 correct.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X < = 8) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 8 ) * 0.5^8 * ( 1- 0.5 ) ^12 + ( 20 7 ) * 0.5^7 * ( 1- 0.5 ) ^13 + ( 20 6 ) * 0.5^6 * ( 1- 0.5 ) ^14 + ( 20 5 ) * 0.5^5 * ( 1- 0.5 ) ^15 + ( 20 4 ) * 0.5^4 * ( 1- 0.5 ) ^16 + ( 20 3 ) * 0.5^3 * ( 1- 0.5 ) ^17 + ( 20 2 ) * 0.5^2 * ( 1- 0.5 ) ^18 + ( 20 1 ) * 0.5^1 * ( 1- 0.5 ) ^19 + ( 20 0 ) * 0.5^0 * ( 1- 0.5 ) ^20
= 0.2517