$A$ box contains $6$ nails and $10$ nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random,what is the probability that it is rusted or is a nail?

Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is $\frac{1}{4}$ and the probability that the second student gets qualified in the same exam is $\frac{2}{5}$,then the probability that at least one of them gets qualified in that exam is

The probability that a number selected at random from the set of numbers $(1, 2, 3, \ldots, 100)$ is a cube,is

Let $A$ and $B$ be events in a sample space $S$ such that $P(A)=0.5, P(B)=0.4$ and $P(A \cup B)=0.6$. Observe the following lists. The correct match of List $I$ from List $II$ is:

List $I$	List $II$
$(i) \ P(A \cap B)$	$(1) \ 0.4$
$(ii) \ P(A \cap \bar{B})$	$(2) \ 0.2$
$(iii) \ P(\bar{A} \cap B)$	$(3) \ 0.3$
$(iv) \ P(\bar{A} \cap \bar{B})$	$(4) \ 0.1$

Two dice are thrown simultaneously. What is the probability of obtaining a multiple of $2$ on one of them and a multiple of $3$ on the other?

$A$ fair coin with $1$ marked on one face and $6$ on the other and a fair die are both tossed. Find the probability that the sum of the numbers that turn up is $12$.