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.

Hint: We will be using the concepts of probability to solve the problem. We will be using the concepts of compound probability to further simplify the problem.

Complete step-by-step solution -
Now we have been given that two buses A and B are scheduled to arrive at a town central bus station at noon.
The probability that the bus A is late
$P\left( A \right)=\dfrac{1}{5}$ ……………………. (1)
The probability that bus B is late
$P\left( B \right)=\dfrac{7}{25}$ …………………. (2)
Probability that B is late given that A is late
$\dfrac{9}{10}=P\left( \dfrac{B}{A} \right)$ ……………….. (3)
Now, we have to find the probability that wither bus will be late on a particular day is $P\left( \overline{A}\cap \overline{B} \right)$ .
Now, we know that $P\left( \dfrac{B}{A} \right)=\dfrac{P\left( B\cap A \right)}{P\left( A \right)}$ .
Now, we will substitute the value of $P\left( {B}/{A}\; \right)$ and $P\left( A \right)$ from (1) and (3) so that we have
$\dfrac{1}{5}\times \dfrac{9}{10}=P\left( B\cap A \right)$ .
$P\left( B\cap A \right)=\dfrac{9}{50}$ ……………… (4)
Now we know that $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$ ,
So, we will substitute the value of $P\left( A\cap B \right)$ , $P\left( A \right)$ and $P\left( B \right)$.
\begin{align} & P\left( A\cup B \right)=\dfrac{1}{5}+\dfrac{7}{25}-\dfrac{9}{50} \\ & =\dfrac{12}{25}-\dfrac{9}{50} \\ & =\dfrac{24-9}{50} \\ & =\dfrac{15}{50} \\ & P\left( A\cup B \right)=\dfrac{3}{10} \\ \end{align}
Now, we have to find the value of $P\left( \overline{A}\cap \overline{B} \right)$ we know that,
$P\left( \overline{A}\cap \overline{B} \right)=P\left( \overline{A\cup B} \right)$ by demerger’s law.
$P\left( \overline{A}\cap \overline{B} \right)=1-P\left( A\cup B \right)$ .
Now, we will substitute the value of $P\left( A\cup B \right)$ .therefore,
\begin{align} & P\left( \overline{A}\cap \overline{B} \right)=1-\dfrac{3}{10} \\ & =\dfrac{7}{10} \\ \end{align}
So, the probability that neither bus will be late $=\dfrac{7}{10}$.
Now, we have to find the probability that bus A is late given that bus B is late is $P\left( \dfrac{A}{B} \right)$ .
Now, we know that $P\left( \dfrac{A}{B} \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)}$.
Now, we will substitute the value from (2) and (4)
\begin{align} & =\dfrac{9}{50}\times \dfrac{25}{7} \\ & =\dfrac{9}{14} \\ & =\dfrac{18}{28} \\ \end{align}
Hence the correct option is (B).

Note: To solve these types of question we must have a basic understanding of probability also it is important to remember that
\begin{align} & P\left( \dfrac{A}{B} \right)=\dfrac{P\left( A\cap B \right)}{P\left( B \right)} \\ & P\left( \overline{A} \right)=1-P\left( A \right) \\ & P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right). \\ \end{align}