Answer to Question #145912 in Statistics and Probability for rwan alqmash

Question

Ninety percent of flights depart on time. Eighty percent of flights arrive on time.
Seventy-five percent of flights depart on time and arrive on time. What is the
probability that
a. A flight will arrive on time if it departed on time.
b. A flight will arrive on time if it didn't depart on time.
c. Are the events departing on time and arriving on time independent?

Answer 1

Let D represent departing on time and A represent arriving on time.

$P(D)=0.9$

$P(A)=0.8$

$P(D\cap A)=0.75$

a. P(A|D)

$P(A|D)=\frac{P(A\cap D)}{P(D)}=\frac{0.75}{0.9}=0.833$

b. $P(A|D^c)$

$P(D^c)=1-0.9=0.1$

$P(A\cap D^c)=0.1-(1-0.9-0.8+0.75)=0.05$

$P(A|D^c)=\frac{P(A\cap D^c)}{P(D^c)}=\frac{0.05}{0.1}=0.5$

c. The events departing on time and arriving on time are not independent. $P(A\cap D)=0.75\ne P(A)×P(D)=0.72$