A person goes to office either by car, scooter, bus or train, the probabilities of which being

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A person goes to office either by car, scooter, bus or train, the probabilities 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 what is the probability that he travelled by a car.

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Let the events E1, E2, E3, E4, and A be defined as follows:

E1: Event that the person goes to the office by car

E2: Event that the person goes to the office by scooter

E3: Event that the person goes to the office by bus

E4: Event that the person goes to the office by train.

A: Event that the person reaches the office in time

Then, P(E1) = $\frac{1}{7}$ ,P(E2) = $\frac{3}{7}$ , P(E3) = $\frac{2}{7}$ , P(E4) = $\frac{1}{7}$

P(A/E1) = P(Person reaches office in time if he goes by car)

= 1 – P(Person reaches office late if he goes by car)

$1- \frac{2}{9}​​​​=\frac{7}{9}$

= P(A/E2) = P(Person reaches office in time if he goes by scooter)

= 1 – P(Person reaches office late if he goes by scooter) =

$1- \frac{1}{9} =\frac{8}{9}$

P(A/E3) = P(Person reaches office in time if he goes by bus)

= 1 – P(Person reaches office late if he goes by bus)

$1-\frac{4}{9}=\frac{5}{9}$

P(A/E4) = P(person reaches office in time if he goes by train)

= 1 – P(person reaches office late if he goes by train)

$1-\frac{1}{9}=\frac{8}{9}$

Now, P(person travelled by car if he reached office in time)

P(E1/A) = $\frac {P(E_1)\times P(A/E_1)}{P(E_1)\times P(A/E_1)+ P(E_2)\times P(A/E_2)+P(E_3)\times P(A/E_3)+P(E_4)\times P(A/E_4)}$   ....  Baye's Theorem

$\frac{\frac{1}{7}\times \frac{7}{9}}{\frac{1}{7}\times \frac{7}{9}+ \frac{3}{7}\times \frac{8}{9}+ \frac{2}{7}\times \frac{5}{9}+ \frac{1}{7}\times \frac{8}{9}}$ = $\frac{\frac{7}{63}}{\frac{7}{63}+\frac{24}{63}+\frac{10}{63}+\frac{8}{63}}$ = $\frac{\frac{7}{63}}{\frac{49}{63}}$ = $\frac{7}{49} =\frac{1}{7}$