Total Probability Theorem

Q. A bag contains $n+1$ coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that the toss results in heads is $\frac{7}{12}$, then the value of $n$ is

Q. An urn contains $5$ red and $2$ green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is :

1. $\frac{27}{49}$
2. $\frac{21}{49}$
3. $\frac{26}{49}$
4. $\frac{32}{49}$

Q. If $A$ and $B$ are two events such that $P\left(A\right)=\frac{3}{4}$ and $P\left(B\right)=\frac{5}{8}$, then

1. $P(A\cup B)\ge \frac{3}{4}$
2. $P(A^{\prime }\cup B)\ge \frac{1}{4}$
3. $\frac{3}{8}\le P(A\cup B)\le \frac{5}{8}$
4. $\frac{3}{8}\le P(A\cap B)\le \frac{5}{8}$