Suppose twelve independent identical laboratory experiments

Suppose twelve independent identical laboratory experiments are to be undertaken extremely sensitive to the environmental conditions, and there is only a probability 0.4 that will be completed successfully. Find the probability of a) exactly five successful experiments occures b) at least six experiments are fail c) less than four experiments are success

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 = 5 ) = ( 12 5 ) * ( 0.4^5) * ( 1 - 0.4 )^7
= 0.227

b)

P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 12 5 ) * 0.6^5 * ( 1- 0.6 ) ^7 + ( 12 4 ) * 0.6^4 * ( 1- 0.6 ) ^8 + ( 12 3 ) * 0.6^3 * ( 1- 0.6 ) ^9 + ( 12 2 ) * 0.6^2 * ( 1- 0.6 ) ^10 + ( 12 1 ) * 0.6^1 * ( 1- 0.6 ) ^11 + ( 12 0 ) * 0.6^0 * ( 1- 0.6 ) ^12
= 0.1582
P( X > = 6 ) = 1 - P( X < 6) = 0.8418

c)
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 12 3 ) * 0.4^3 * ( 1- 0.4 ) ^9 + ( 12 2 ) * 0.4^2 * ( 1- 0.4 ) ^10 + ( 12 1 ) * 0.4^1 * ( 1- 0.4 ) ^11 + ( 12 0 ) * 0.4^0 * ( 1- 0.4 ) ^12
= 0.2253