Question

Two buses A and B are scheduled to arrive at a town central bus station at noon. The probability that bus A will be late is 1/5. The probability that bus B will be late is 7/25. The probability that the bus B is late given that bus A is late is 9/10. Then the probabilities
(i) neither bus will be late on a particular day and
(ii) bus A is late given that bus B is late, are respectively

• A. 2/25 and 12/28
• B. 18/25 and 22/28
• C. 7/10 and 18/28
• D. 12/25 and 2/28

Answer 1

Answer: (C)

7/10 and 18/28

P(A) ; denote the probability that bus A will be late.
P(B) ; denote the probability that bus B will be late.
$P\left(A\right)=\frac{1}{5},P\left(B\right)=\frac{7}{25}$
$P\left(\frac{B}{A}\right)=\frac{9}{10}\Rightarrow P(A\cap B)=\frac{9}{50}$
(i) $P(\overline{A}\cap \overline{B})=1-P(A\cup B)$
$=1-(\frac{1}{5}+\frac{7}{25}-\frac{9}{50})=\frac{35}{50}=\frac{7}{10}$
$\left(ii\right)P\left(\frac{A}{B}\right)=\frac{P\left(A\right)\cdot P\left(\frac{B}{A}\right)}{P\left(B\right)}=\frac{\frac{1}{5}\times \frac{9}{10}}{\frac{7}{25}}=\frac{18}{28}$