$A$ coin is tossed until a head appears or it has been tossed thrice. Given that a head does not appear on the first toss,what is the probability that the coin is tossed thrice?

Consider the following statements.
Statement $(I)$: If $E$ and $F$ are two independent events,then $E^{\prime}$ and $F^{\prime}$ are also independent.
Statement $(II)$: Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent.
Which of the following is correct?

Bag $A$ contains $9$ white and $8$ black balls, while bag $B$ contains $6$ white and $4$ black balls. One ball is randomly picked up from bag $B$ and mixed with the balls in bag $A$. Then a ball is randomly drawn from bag $A$. If the probability that the ball drawn is white is $p/q$ (where $gcd(p,q)=1$), then $p+q$ is equal to:

Two numbers are selected at random from the set $\{1, 2, 3, \ldots, 13\}$. If the sum of the selected numbers is even, what is the probability that both the numbers are odd?

If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$,and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$,then the value of $\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}$ is:

$A$ candidate takes three tests in succession and the probability of passing the first test is $p$. The probability of passing each succeeding test is $p$ if he passes the preceding one, and $\frac{p}{2}$ if he fails the preceding one. The candidate is selected if he passes at least two tests. The probability that the candidate is selected is: