A person goes to office either by car, scooter, bus or train, the probability of which being $${1 \over 7},{3 \over 7},{2 \over 7}$$ and $${1 \over 7}$$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $${2 \over 9},{1 \over 9},{4 \over 9}$$ and $${1 \over 9}$$ respectively. Given that he reached office in time, then what is the probability that he travelled by a car.

A box contains $$12$$ red and $$6$$ white balls. Balls are drawn from the box one at a time without replacement. If in $$6$$ draws there are at least $$4$$ white balls, find the probability that exactly one white is drawn in the next two draws. (binomial coefficients can be left as such)

$$A$$ and $$B$$ are two independent events. $$C$$ is even in which exactly one of $$A$$ or $$B$$ occurs. Prove that $$P\left( C \right) \ge P\left( {A \cup B} \right)P\left( {\overline A \cap \overline B } \right)$$

$$A$$ is targeting to $$B, B$$ and $$C$$ are targeting to $$A.$$ Probability of hitting the target by $$A,B$$ and $$C$$ are $${2 \over 3},{1 \over 2}$$ and $${1 \over 3}$$ respectively. If $$A$$ is hit then find the probability that $$B$$ hits the target and $$C$$ does not.