2.1 & 2.2 Probability

Question 1

Probability Experiment

Answer 1

can be repeated indefinitely with a fixed set of outcomes

Question 2

Probability

Answer 2

measure of how likely a given event will happen

*random in short term, predictable in long term

Question 3

Law of Large Numbers

Answer 3

increase #repetitions, increases outcome approaching probability

Question 4

Sample Space

Answer 4

S = fixed set of all possible outcomes
S={e1, e2, e3,…}
S=(heads, tails)
S={1,2,3,4,5,6}

Question 5

Event

Answer 5

some COMBINATION of outcomes

doesn’t have to be just one thing

Question 6

Simple Event

Answer 6

exactly one outcome

rolling a 1: e={1}

Question 7

Compound Event

Answer 7

more than one outcomes

rolling an even number: e={2,4,6}

Question 8

Complement

Answer 8

A ⊆ S = event
A’ = all simple events in S not in A
A’ = S \ A (set S minus set A)

Question 9

Union

Answer 9

A, B ⊆ S
set of simple events in A, B, or both A and B
A ⋃ B = {e ∈ S | e ∈ A or e ∈ B}

Question 10

Intersection

Answer 10

A, B ⊆ S
set of simple events in both A and B
A ⋂ B = {e ∈ S | e ∈ A and e ∈ B}

Question 11

null event

Answer 11

event contains no outcomes

Ø = { }

Question 12

Disjoint

Answer 12

two events A, B ∈ S = disjoint (mutually exclusive) if A ⋂ B = Ø (nothing in common)
A={2,3,4} and B={1}

Question 13

P(A)

Answer 13

the probability of A
A={"head"}, B={"tails"}
P(A)= .5 = 50%
P(A ⋃ B) =  1 = 100%
P(A ⋂ B) =  P(Ø) = 0 = 0%

Question 14

Proposition (Axioms of probability)

Answer 14

Axiom 1: For any event A, P(A) >= 0
Axiom 2: P(S) = 1
Axiom 3: if A1, A2, A3,…is any finite or infinite collection of disjoing events, then
P(A1⋃A2⋃A3…) = Σ P(Ai)

Question 15

Proposition

Answer 15

For any event A: P(A) <=1

Question 16

Probability Model

Answer 16

list of outcomes with their probabilities
P(heads) = 1/2
P(tails) = 1/2
(rolling a die) P(1) -> P(6) = 1/6

Question 17

When is an event Impossible, Unlikely, and certain?

Answer 17

Impossible: P(E) = 0
Unlikely: P(E) < = 5%
Certain: P(E) = 1

Question 18

the Classical Method

Answer 18

when all outcomes of probability experiment are equally likely
P(E) = #ways outcome can occur / total # possible outcomes

Question 19

Proposition (Complement Rule)

Answer 19

for an event A
P(A) + P(A') = 1 <=> P(A') = 1-P(A)
ex. P({2,4,5,6}) = 2/3
complement rule:
P({1,3}) = 1 - (2/3) = 1/3

Question 20

Proposition (Axiom 3)

Answer 20

if 2 events E and F are disjoint, then
P(E⋃F) = P(E) + P(F)
ex. rolling a die define 2 events
E={2,4} and F = {5,6} then
P(E⋃F) = (2/6) + (2/6) = 4/6 = 0.667

Question 21

Axiom 3 when two elements are NOT disjoint

Answer 21

P(red card) = 26/52 = 1/2
P(king) = 4/52 = 1/13
P(red card or king) = 1/2 + 1/13 = 15/26? NO, king of hearts and king diamonds = double counted
-general addition rule: subt. common probability

Question 22

General Addition Rule

Answer 22

subtract off common probability
P(A⋃B) = P(A) + P(B) - P(A⋂B)
if A and B are disjoint, then P(A⋂B) = 0

Question 23

For any three events A, B and C

Answer 23

P(A⋃B⋃C) = P(A) + P(B) + P(C)
−P(A∩B)−P(A∩C)−P(B ∩C)
+ P(A∩B ∩C)
subt. pair-wise intersections, add triple
similar formulas exist for any number of events