The odds against a certain event are $9$ to $5$ and the odds in favour of another event are $5$ to $3$. If the two events are independent, then find the probability that at least one of them will occur.

The odds against a man who is $45$ years old, living till he is $70$ are $7:5$ and the odds against his wife who is now $36$, living till she is $61$ are $5: 3$. Find the probability that the couple will be alive $25$ years hence

The odds against a man who is $45$ years old, living till he is $70$ are $7:5$ and the odds against his wife who is now $36$, living till she is $61$ are $5: 3$. Find the probability that at least one of them will be alive $25$ years hence

A machine operates only when all of its three components function. The probabilities of the failures of the first second and third components are $0.14, 0.10$ and $0.05$ respectively. What is the probability that the machine will fail?

A problem in Mathematics is given to three students whose chances of solving it correctly are $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$ respectively. What is the probability that only one of them solves it correctly?