The 3 year survival rate for a certain group of cancer patie

The 3 year survival rate for a certain group of cancer patients is known to be 0.60. What is the probability that ina group of 10 patients more than 5 patients survive at least 3 years

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 < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 4 ) * 0.6^4 * ( 1- 0.6 ) ^6 + ( 10 3 ) * 0.6^3 * ( 1- 0.6 ) ^7 + ( 10 2 ) * 0.6^2 * ( 1- 0.6 ) ^8 + ( 10 1 ) * 0.6^1 * ( 1- 0.6 ) ^9 + ( 10 0 ) * 0.6^0 * ( 1- 0.6 ) ^10   
= 0.1662

P( X > = 5 ) = 1 - P( X < 5) = 0.8338