lecture 10

Question 1

subjective probability

Answer 1

probability was the degree of belief in a statement

do not work with this one in stats115

Question 2

Objective probability

Answer 2

probability as a measure of the relative frequency ‘in the long run’ of an outcome

work with this one in stat115

Question 3

Experiment

Answer 3

an experiment is a process by which observations/measurements are obtained, e.g.:

• throwing a fair die,
• taking measurements of blood pressure,
• asking people which ethnic groups they identify with.

Question 4

Event

Answer 4

An event is the outcome of an experiment, e.g.: - getting a 6,
- a systolic blood pressure measure of 120 mmHg, - identifying as M ̄aori

needs to be clear, one possible outcome of the experiment

Question 5

Sample space

Answer 5

The sample space is the set of all possible outcomes of an experiment, e.g.:
- in the case of a fair die, {1,2,3,4,5,6}.

Question 6

Probability of an event A is

Answer 6

Pr(A) = number of experiments resulting in A/ large number of repetitions = nA/N

nA is the number of repetitions that involved A being satisfied

repeating gives more realisations

Question 7

Complementary events

Answer 7

Two events are complementary if all outcomes are either one of the two events, e.g head or tail on fair coin.
A ̄ is called the complement of A and

Pr(A) + Pr(A ̄) = 1

Question 8

Mutually exclusive events

Answer 8

There is no intersection between the two events. - not able to see both events in one go
In other words, events are mutually exclusive if they cannot both occur.
Back to coin flipping: getting heads and tails on the same toss. Which of the following are mutually exclusive?
Never smoked and ever smoked - mutually exclusive
Maori and European ethnicity - not mutually exclusive
Cardiovascular disease and diabetes - not mutually exclusive

Question 9

Conditions for a valid probability

Answer 9

1 - Probabilities take values from 0 to 1.
2 - The sum of the probabilities over all possible events is 1.
- In other words, the total probability for all possible outcomes of a random circumstance is equal to 1 (as long as these events are mutually exclusive).

Question 10

if the event A cannot happen then

Answer 10

Pr(A) = 0

The event that the mouse we trap is a male and pregnant has a probability Pr(A) = 0 of occurring because this is impossible.

Question 11

if the event A is certain to happen then

Answer 11

Pr(A) = 1

The event that the mouse we trap is either male or not a male has a probability Pr(A) = 1 of occurring because it will definitely be one or the other.

Question 12

The event either A or B occurs is:

Answer 12

AorB =AunionB =A∪B

Question 13

The event both A and B occurs is:

Answer 13

AandB =AintersectB =A∩B

satisfy both events at a given time

Question 14

Conditional events

Answer 14

Two events are conditional if the probability of one event changes depending on the outcome of another event.

Important not to interpret conditional results as unconditional.

What is the probability of buying ice cream? Hot day = High, Cold day = Low

We write conditional probabilities as: Pr(A | B)
(read as ‘the probability of A given B’)

what is the probability of event 1 given the probability of a previous event 2

order matters because what we are conditioning on matters

Question 15

Multiplication rule

Answer 15

used with respect to intersections

pr(A) probability of what we are conditioning on
Pr (B | A) - conditional probability

Pr(A and B) = Pr(A ∩ B) = Pr(A) Pr(B|A)

Question 16

Addition rule

Answer 16

Pr(A or B) = Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)

used with respect to unions

two events mutually exclusive probability = 0

Question 17

Mutually exclusive events and addition rule

Answer 17

events are mutually exclusive if they cannot both occur. This means there is no intersection between the two events.

For example: for a single flip of a coin, the events A = heads and B = tails cannot both occur.
In this case, Pr(A and B) = Pr(A ∩ B) = 0

Using the addition rule:
Pr(AorB)=Pr(A ∪ B)
= Pr(A) + Pr(B) − Pr(A ∩ B)
= Pr(A) + Pr(B)

Question 18

Independent events and the multiplication rule

Answer 18

Events are independent when the occurrence of one event does not affect the outcome of another event.
For example: suppose we flip a coin 2 times in a row.
Define event A as heads on first flip and event B as heads on second flip.
In this case, Pr(B|A) = Pr(B)

Using the multiplication rule:
Pr(A and B) = Pr(A ∩ B)
= Pr(A)Pr(B|A)
= Pr(A)Pr(B)

For independent events - end up with product of the 2 isolated variables