The probability that a mechanic makes an error while using a machine on the $n$th day is given by $P(E_n) = \frac{1}{2^n}$. If he has operated the machine for $4$ days, the probability that he has not made a mistake on $3$ of the $4$ days is:

$S$ is the sample space and $A, B$ are two events of a random experiment. Match the items of List-$A$ with the items of List-$B$.

List-$A$	List-$B$
$(I)$ $A, B$ are mutually exclusive events	$(i)$ $P(A \cap B) = P(B) - P(\bar{A})$
$(II)$ $A, B$ are independent events	$(ii)$ $P(A) \leq P(B)$
$(III)$ $A \cap B = A$	$(iii)$ $P(\frac{\bar{A}}{B}) = 1 - P(A)$
$(IV)$ $A \cup B = S$	$(iv)$ $P(A \cup B) = P(A) + P(B)$
	$(v)$ $P(A) + P(B) = 2$

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that one of them is black and the other is red.

If two unbiased six-faced dice are thrown simultaneously until a sum of either $7$ or $11$ occurs,then the probability that $7$ comes before $11$ is

$A$ random variable $X$ has the following probability distribution:
| $X=x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $P(X=x)$ | $0.15$ | $0.23$ | $0.12$ | $0.20$ | $0.08$ | $0.10$ | $0.05$ | $0.07$ |
For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 5\}$,find $P(E \cup F)$.

In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of $52$ playing cards by a person $B$. They win the game if $A$ gets a prime score as the sum of the numbers appearing on both the dice and $B$ gets a face card and a card having a prime number. Then the probability that both $A$ and $B$ win is: