Probability, permutations and combinations

The probability of events happening together or separately, and how many different ways a selection can be made. What they are for is explained in Chapter 4.

Probability of events

Example:

A bag has 5 red, 6 yellow and 7 green marbles. Three draws are made, returning the marble each time: what is the probability of red, then yellow, then green?

How do the events combine?All together (and): multiplyAny one (or): add

Event	Favourable outcomes	All outcomes	Probability	Delete
			0.2778
			0.3333
			0.3889

Result

Probability that all occur 0.036

If the events do not affect each other (independent), multiply the probabilities.
$P(A \text{ and } B) = P(A) \cdot P(B)$
$P = \dfrac{5}{18} \cdot \dfrac{6}{18} \cdot \dfrac{7}{18} = \dfrac{210}{5832} = \dfrac{35}{972} = 0.036008$

Counting

Example:

In how many ways can a 3-person jury be chosen from 10 professors? Jury members have no order.

Does the order of the chosen items matter?Yes (permutation)No (combination)Arrange all (n!)

Total (n)Chosen (r)

Result

Combination 120

Order does not matter: the same people form one group whatever order they are chosen in. Divide the number of permutations by the number of ways r people can be ordered among themselves (r!).
${}_nC_r = \dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!} = \dfrac{{}_nP_r}{r!}$
${}_{10}C_{3} = \dfrac{10!}{3!\,(10-3)!} = \dfrac{10!}{3!\cdot 7!}$
$= \dfrac{720}{3!} = \dfrac{720}{6} = 120$