Chapter 4: Probability

Question 1

sample space

Answer 1

set of all possible outcomes, S
varies depending on how you organize outcomes
e.g. sum of dice rolled: does order matter or not? changes S

Question 2

cardinality

Answer 2

of all elements in the set

Question 3

event

Answer 3

subset of a sample space

Question 4

set notation

Answer 4

A’ = complement of A
AUB = Union. Set of all outcomes in A or B or both
AnB = Intersection. All outcomes that in A and B
{ } = Null set, empty set. Event with no possible outcomes

Question 5

mutually exclusive, collectively exhaustive

Answer 5

ME: AnB = { }, A and B are disjoint. “did you double count?”
CE: “did you miss anything?”

Question 6

classic definitions of probability

Answer 6

1. P(A) = n(A)/n(S)
   cardinality of A over total number of events in sample space
2. P(A) = lim(N→100) n/N
   experiment runs N times and # of times event A occurs is n

Question 7

3 axioms of probability

Answer 7

1. 0 ≤ P(A) ≤ 1 (probabilities have to be between 0 and 1)
2. P(S) = 1 (probably of sample space is always 100%)
3. If E1, E2 … En are ME events
   → P (E1 U E2 U … U En) = P(E1) + P(E2) + …. + P(E3)

Question 8

probability of the complement

Answer 8

P(A’) = 1–P(A)

Question 9

Addition rule

Answer 9

P(AUB) = P(A) + P(B) – P(AnB)
- can’t just add probabilities because that would be double counting
- extends to 2+ events

Question 10

probabilities where order matters and doesn’t matter, replacement or without replacement

Answer 10

ORDER MATTERS + REPLACEMENT
- n^r

ORDER MATTERS + NO REPLACEMENT
- factorial
- when you want to select, pick function

ODER DOESN’T MATTER + REPLACEMENT
- don’t worry about it

ORDER DOESN’T MATTER + NO REPLACEMENT
- choose function

Question 11

conditional probability

Answer 11

P(A|B) = P(AnB)/P(B)

Question 12

what does it mean for events to be independant?

Answer 12

two events are independent only if

P(A|B) = A

so that the occurrence of B does not change P(A)

Question 13

general multiplication rule

Answer 13

rearrange conditional probability equation to get

P(AnB) = P(B) *P(A|B)
P(AnB) = P(A) * P(B|A)

Question 14

multiplication rule for independent events/test for independence

Answer 14

If A and B are independent,
P(AnB) = P(B) *P(A)

Question 15

Baye’s formula