Partial Permutations

Find the number of 7-character (capital letter or digit) license plates possible if no character can repeat and: a) there are no further restrictions, b) the first 3 characters are letters and the last 4 are numbers, c) letters and numbers alternate, for example A3B9D7Q or 0Z3Q4A9

Combinations

A standard 52-card deck consists of 4 suits and 13 ranks. Find the number of 5-card hands where: a) any hand is allowed (namely the number of different hands) b) all five cards are of same suit c) all four suits are present d) all cards are of distinct ranks

Distribution Types

1) Which of the following sample spaces are uniform?

a) {land, sea} for a randomly point on a globe b) {odd, even} for a random integer from {1,2,. . . ,100} c) {leap year, non-leap year} for a random year before 2019 d) {two he{distance to origin} for a random point in {−3, −1, 1, 3} × {−4, −2, 2, 4} e) lads, two tails, one head and one tail} when flipping two fair coin

Inequalities

.in any uniform probability space: a) ⊇ ⟶ P()≥P() b) P()≥P() ⟶ ⊇ c) ||≥|| ⟶ P()≥P() d) P()≥P() ⟶ ||≥||

Conditional Probability

Three fair coins are sequentially tossed. Find the probability that all are heads if: a) the first is tails b) the first is heads c) at least one is heads.