Module 2: Axioms of Probability

Question 1

what are the three probability axioms?

Answer 1

nonnegativity, additivity and normalization

Question 2

what does the nonnegativity probability axiom mean?

Answer 2

the probability of any event, P(E) is greater than or equal to 0

Question 3

what does the additivity probability mean?

Answer 3

If E1 and E2 are two disjoint (mutually exclusive) events, then the
probability of their union satisfies

P (E1 ∪ E2) = P (E1) + P (E2).

Furthermore, if the sample space has an infinite number of elements and E1,E2, . . .
is a sequence of disjoint events, then the probability of their union satisfies

P (E1 ∪ E2 ∪ . . .) = P (E1) + P (E2) + . . .

Question 4

what does the normalization axiom mean?

Answer 4

With probability 1, the outcome will be a point in the sample
space S:

P (S) = 1.

In other words, the probability of the entire sample space S equals 1

Question 5

T or F: the axioms are provable.

Answer 5

False, they can’t be proved. The entire field of probability doesn’t make sense unless you assume the axioms.

Question 6

what is the Discrete Uniform Probability Law?

Answer 6

If the sample space S consists of n possible outcomes which are equally likely (i.e., all
single-element events have the same probability), then the probability of any event E is
given by

P(E) = number of elements of E/n

Question 7

what sample space is associated with a discrete probability model?

Answer 7

if the sample space has a finite or a countable infinite number of possible outcomes.

Countably infinity meaning it can be listed or counted in a sequence, even though it goes on forever. Or it’s an infinite sequence but you know what comes next in the sequence

Question 8

what is the associated probability model of a sample space with uncoutable infinite number of possible outcomes.

Answer 8

continuous.

uncountable infinite meaning anything measurable.

Ex. No one is actually exact 6 foot tall

Question 9

what are common ways to visualize probabilities?

Answer 9

tree diagrams and two dimensional grids (only for two different outcomes (See Ex 4b)

Question 10

How would a continuous model work in a practical sense?

Answer 10

there are an infinite amt of #’s from [0,1] so that means P(S) = infinity. we have to narrow the range of #s we could guess from.