A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. What is the probability that it lands head up?

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed.

If happens to be head, what is the probability that it is the two-headed coin?

There are three social media groups on a mobile: Group $I$, Group $II$ and Group $III$. The probabilities that Group $I$, Group $II$ and Group $III$ sending the messages on sports are $\frac{2}{5},\frac{1}{2},$ and $\frac{2}{3}$ respectively. The probability of opening the messages by Group $I$, Group $II$ and Group $III$ are $\frac{1}{2},\frac{1}{4}$ and $\frac{1}{4}$ respectively. Randomly one of the messages is opened and found a message on sports. What is the probability that the message was from Group $III$.

If odds in favour of $X$ solving a problem are $4:3$ and odds against $Y$ solving the same problem are $2:3$. Find probability of $X$ solving the problem.

If odds in favour of $X$ solving a problem are $4:3$ and odds against $Y$ solving the same problem are $2:3$. Find probability of $Y$ solving the problem.

The odds against student $X$ solving a statistics problem are $8:6$ and odds in favour of student $Y$ solving the same problem are $14:16$. Find is the chance that the problem will be solved if they try it independently.

The odds against student $X$ solving a statistics problem are $8:6$ and odds in favour of student $Y$ solving the same problem are $14:16$. Find is the chance that neither of them solves the problem.

The odds against a husband who is $60$ years old, living till he is $85$ are $7:5$. The odds against his wife who is now $56$, living till she is $81$ are $5:3$. Find the probability that at least one of them will be alive $25$ years hence.