Question

A person goes to office either by car, scooter, bus or train probability of which being $\frac{1}{7},\frac{3}{7},\frac{2}{7}$ and $\frac{1}{7}$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is

$\frac{2}{9},\frac{1}{9},\frac{4}{9}$ and $\frac{1}{9}$ respectively. Given that he reached office in time, then if the probability that

he travelled by a car is $\frac{1}{p}.$Find $p$

Answer 1

Let C, S, B, T be the events of the person going by car, scooter, bus or train respectively

Given that P(C) = $\frac{1}{7}$, P(S) = $\frac{3}{7}$ ;P(B) = $\frac{2}{7}$ ; P(T) = $\frac{1}{7}$

Let $\overline{L}$be the event of the person reaching the office in time.

$\Rightarrow P\left(\frac{\overline{L}}{C}\right)=\frac{7}{9},P\left(\frac{\overline{L}}{S}\right)=\frac{8}{9},P\left(\frac{\overline{L}}{B}\right)=\frac{5}{9},P\left(\frac{\overline{L}}{T}\right)=\frac{8}{9}$

$\Rightarrow P\left(\frac{C}{\overline{L}}\right)=\frac{P\left(\frac{\overline{L}}{C}\right).P\left(C\right)}{P\left(\overline{L}/C\right)\cdot P\left(C\right)+P\left(\overline{L}/S\right)\cdot P\left(S\right)+P\left(\overline{L}/B\right)\cdot P\left(B\right)+P\left(\overline{L}/T\right)\cdot P\left(T\right)}$

$=\frac{\frac{1}{7}\times \frac{7}{9}}{\frac{1}{7}\times \frac{7}{9}+\frac{3}{7}\times \frac{8}{9}+\frac{2}{7}\times \frac{5}{9}+\frac{8}{9}\times \frac{1}{7}}=\frac{1}{7}=\frac{1}{p}.$

$\therefore p=7$