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.

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?