A fair coin is tossed 20 times and all tosses are independen

A fair coin is tossed 20 times and all tosses are independent. What is the true probability of obtaining exactly 10 heads? Compute and compare the probability when approximated by an appropriate normal distribution. Compute approximate probability of obtaining more than 7 heads with and without continuity correction. Compare these approximations to the exact probability found using pmf of binomial distribution.

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

a)
P( X = 10 ) = ( 20 10 ) * ( 0.5^10) * ( 1 - 0.5 )^10
= 0.1762

b)
P( X < = 7) = 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 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
= 0.1316
P( X > 7) = 1 - P ( X <=7) = 1 -0.1316 = 0.8684