1) A software function generates a random number N digits long. Each digit is determined by randomly selecting a value from 0 through 9.

All ten values are equally likely, and different digits may have the same value. Determine the minimum length N such that there is at least a 50% probability that at least one digit will have the value 0.

2) Six cards are identical except that both sides of two cards are red, both sides of three cards are black, and the sixth card has one red side and one black side. One of the six cards is chosen at random and placed on a table. If the upper side of the chosen card is red, what is the probability that the bottom side is black?

3) The number of database queries made during any interval of time is a Poisson random variable. On average, six queries are made every minute. Determine the probability that three or more queries will be made during an interval of 15 seconds.

4) The joint probability density function of random variables X and Y is

Determine formulas in closed form for the marginal probability distribution function of X for all x, and sketch a graph of this function over the range -1 ≤ x ≤ 2.