Cards from an ordinary deck of 52 playing cards are turned

#18. Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the first card is an ace, or the second a deuce, or the third a three, or …, or the thirteenth a king, or fourteenth an ace, and so on, we say that a match occurs. Note that we do not require that the (13n + 1)th card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

A lake contains 4 distinct types of fish

#18. A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let Y denote the number of fish that need be caught to obtain at least one of each type. (a) Give an interval (a, b) such that P{a ≤ Y ≤ b} ≥ 0.90. (b) Using the one-sided Chebyshev inequality, how many fish need we plan on catching so as to be at least 90 percent certain of obtaining at least one of each type? (Questoin is also included in attachment)

Let X be a nonnegative random variable

#20. Let X be a nonnegative random variable. Prove that... (see attachment)

The problem is from 500 level in undergraduate/graduate math.

In how many ways can the numbers 1,2,3,4,5,6 be written in a row if (a) There are no restrictions on the order of the numbers? (b) 1 dn 2 are next to each other? ... (SEE ATTACHMENT FOR REST)

Dice probability

A random number of dice is rolled. Find the probability that... (SEE ATTACHED)