A government\'s department of transportation reported that in 2009, airline A le

Question

A government's department of transportation reported that in 2009, airline A led all domestic airlines in on-time arrivals for domestic flights, with a rate of 83.7%. Complete parts a through e below.

a. What is the probability that in the next six flights, exactly four flights will be on time?

b. What is the probability that in the next six flights, two or fewer will be on time?

c. What is the probability that in the next six flights, at least four flights will be on time?

d. What are the mean and standard deviation for this distribution?

The standard deviation of the number of on-time arrivals is

Explanation / Answer

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 = 4 ) = ( 6 4 ) * ( 0.837^4) * ( 1 - 0.837 )^2
= 0.1956
b.
P( X < = 2) = P(X=2) + P(X=1) + P(X=0)
= ( 6 2 ) * 0.837^2 * ( 1- 0.837 ) ^4 + ( 6 1 ) * 0.837^1 * ( 1- 0.837 ) ^5 + ( 6 0 ) * 0.837^0 * ( 1- 0.837 ) ^6
= 0.008
c.
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 6 3 ) * 0.837^3 * ( 1- 0.837 ) ^3 + ( 6 2 ) * 0.837^2 * ( 1- 0.837 ) ^4 + ( 6 1 ) * 0.837^1 * ( 1- 0.837 ) ^5 + ( 6 0 ) * 0.837^0 * ( 1- 0.837 ) ^6
= 0.0588
P( X > = 4 ) = 1 - P( X < 4) = 0.9412
d.
Mean ( np ) =6 * 0.837 = 5.022
Standard Deviation ( npq )= 6*0.837*0.163 = 0.9048

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