The speed with which utility companies can resolve problems

The speed with which utility companies can resolve problems is very important. GTC, the Georgetown Telephone Company, reports it can resolve customer problems the same day they are reported in 76% of the cases. Suppose the 14 cases reported today are representative of all complaints.

What is the standard deviation?

What is the probability 8 of the problems can be resolved today?

What is the probability 8 or 9 of the problems can be resolved today?

What is the probability more than 9 of the problems can be resolved today?

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)
Standard Deviation ( npq )= 14*0.76*0.24 = 1.598

b)
P( X = 8 ) = ( 14 8 ) * ( 0.76^8) * ( 1 - 0.76 )^6
= 0.0639
c)
P( X = 9 ) = ( 14 9 ) * ( 0.76^9) * ( 1 - 0.76 )^5
= 0.1348

P( X = 8 OR X = 9) = 0.0639 + 0.1348 = 0.1987
d)

P( X < 9) = 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)   
= ( 14 8 ) * 0.76^8 * ( 1- 0.76 ) ^6 + ( 14 7 ) * 0.76^7 * ( 1- 0.76 ) ^7 + ( 14 6 ) * 0.76^6 * ( 1- 0.76 ) ^8 + ( 14 5 ) * 0.76^5 * ( 1- 0.76 ) ^9 + ( 14 4 ) * 0.76^4 * ( 1- 0.76 ) ^10 + ( 14 3 ) * 0.76^3 * ( 1- 0.76 ) ^11 + ( 14 2 ) * 0.76^2 * ( 1- 0.76 ) ^12 + ( 14 1 ) * 0.76^1 * ( 1- 0.76 ) ^13 + ( 14 0 ) * 0.76^0 * ( 1- 0.76 ) ^14

= 0.0949
P( X > = 9 ) = 1 - P( X < 9) = 0.9051