If $$p$$ and $$q$$ are chosen randomly from the set $$\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},$$ with replacement, determine the probability that the roots of the equation $${x^2} + px + q = 0$$ are real.

In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, $$3$$ in the front and $$4$$ at the back? How many seating arrangements are possible if $$3$$ girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of $$3$$ girls sitting together in a back row on adjacent seats?

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $$2, 3,4,.....12$$ is picked and the number on the card is noted. What is the probability that the noted number is either $$7$$ or $$8$$?

Numbers are selected at random, one at a time, from the two- digit numbers $$00, 01, 02 ......, 99$$ with replacement. An event $$E$$ occurs if only if the product of the two digits of a selected number is $$18$$. If four numbers are selected, find probability that the event $$E$$ occurs at least $$3$$ times.