Conditional Probability Questions

A die is thrown twice and the sum of the dots on the two faces is observed to be 7. The conditional probability that the number 2 has appeared at least once is:

If $P(A)=0.4$, $P(B)=0.8$, and $P(B|A)=0.6$, then $P(A\cup B)$ is equal to:

Below is a figure indicating a sample space $S$ and the probability of occurrences of events $A$ and $B$. Which colored section of the portion will indicate the probability of events $B$ and $A$ occurring simultaneously?

Given $P(H)=0.3$, $P(R)=0.4$, and $P(H|R)=0.2$. Find $P(R\cap H)$.

In a job placement event, there are $1000$ aspirants attending a company interview, out of which $430$ are females. It is known that $10\%$ of the females are graduates. Determine the probability that an aspirant selected randomly is a graduate given that the selected aspirant is a female?

A pair of fair dice is thrown. What is the probability of getting a six on both, given at least one die gets six?

A die is thrown 3 times. Let Event A be "4 on the 3rd throw" and Event B be "6 on the 1st and 5 on the 2nd throw". What is the probability of A given that B has already occurred?

A player tosses three coins. If the first toss is a Head, find the probability that there are two consecutive Heads among the three coins.

A family has two children. If the first child is a boy, find the probability that the other child is also a boy.