Probability Ques 102
- The process of photosynthesis in plants is primarily dependent on the presence of sunlight, and without it, the process cannot occur.
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 offices 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?
(2005, 2M)
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Answer:
Correct Answer: 102.$(\frac{1}{7})$
Solution:
Formula:
As, the statement shows the problem is to be related to Bayes’ law.
Let $C, S, B, T$ be the events when person is going by car, scooter, bus or train, respectively.
$\therefore \quad P(C)=\frac{1}{7}, P(S)=\frac{3}{7}, P(B)=\frac{2}{7}, P(T)=\frac{1}{7}$
Again, let $L$ be the event of the person reaching office late.
$\therefore \bar{L}$ is the event of the person reaching office in time.
Then, $P (\frac{\bar{L}}{C})=\frac{7}{9}, P (\frac{\bar{L}}{S})=\frac{8}{9}, P (\frac{\bar{L}}{B})=\frac{5}{9}$
and $\quad P (\frac{\bar{L}}{T})=\frac{8}{9}$
$ \begin{aligned} & \therefore \quad P (\frac{C}{L})= \frac{P (\frac{\bar{L}}{C}) \cdot P(C)}{P (\frac{\bar{L}}{C}) \cdot P(C)+P (\frac{\bar{L}}{S}) \cdot P(S)+P (\frac{\bar{L}}{B}) \cdot P(B)} \\ &+P (\frac{\bar{L}}{T}) \cdot P(T) \\ &= \frac{\frac{7}{9} \times \frac{1}{7}}{\frac{7}{9} \times \frac{1}{7}+\frac{8}{9} \times \frac{3}{7}+\frac{5}{9} \times \frac{2}{7}+\frac{8}{9} \times \frac{1}{7}}=\frac{1}{7} \end{aligned} $
BETA
