sample space and events

Sample Space and Events Section 2.1

An experiment: is any action, process or phenomenon whose outcome is subject to uncertainty.

An outcome: is a result of an experiment.

Each run of an experiment results in only one outcome!

A sample space: is the set of all possible outcomes, S, of an experiment.

An event: is a subset of the sample space. An event occurs when one of the outcomes that belong to it occurs.

A simple (or elementary) event: is a subset of the sample space that has only one outcome.

Still in Ch2:

2.2 Axioms, interpretations and properties of probability

Axioms, interpretation and properties Section 2.2

Probability: is a measure of the chance that an event might occur (before it really occurs).

Probability (frequentist): the limiting relative frequency of the occurrence of an event when running the associated experiment over and over again for a very long time, which gives us an indication about the chance of observing that event again when running the experiment one more time (followed in this book)

Probability (Bayesian): A measure of belief (objective or subjective) of the chance that an event might occur.

Philosophically:


Example: Flipping a fair coin.

Frequentist: If we flip a physically balanced (fair) coin over and over again for a long time then the proportion of times that one will observe a head will be ½ of the time. This assumes that air friction and other factors are either controlled or negligible.

Bayesian: If a coin is physically balanced (fair) and if the effect of air friction and other factors that might affect its orientation when landing are all either negligible, or controlled for, then the chance that a head is observed when that coin is flipped is ½.


Mathematically

Probability: is a function that uses events as an input and results in a real number between 0 and 1 as an output that describes the chance that an event might occur.

Because it uses events as input and because events are sets (they are subsets of the sample space) a probability function is said to be a set function.


A probability function must obey some rules for it to make sense mathematically and logically. These rules are called axioms

1) For any event A,

Axioms (rules) of probability:

2) Probability of observing the sure event, S, if you run the experiment is:

3) If A1, A2, A3, …is an infinite collection of disjoint events (i.e. for any i and j where ), then


Axioms (rules) of probability:

3) We can also show that if A1, A2, A3, … An is a finite collection of disjoint events (i.e. for any i and j where ), then



Examples:

1) Studying the blood types of a randomly sampled individual:

Experiment:

Set of possible outcomes (sample space), S:

Choose an individual at random and observe his/her blood type.

Goal: Observe an individual’s blood type

{O, A, B, AB}

Events (subsets of S): (16 of them)


Examples:


Let E be the simple event {O}, is the following correct?

P(E) = -0.5


Examples:


Let E be the simple event {O}, C be the event {A, AB} and D the event {O, A, AB}, is the following correct?

P(E) = 0.2

P(C) = 0.3

P(D) = 0.4

Let E1 be the simple event {O}, E2 be {A}, E3 be {B} and E4 be {AB}, is the following correct?

Section 2.2

Examples:


P(E1) = 0.2

P(E2) = 0.3

P(E3) = 0.4

P(E4) = 0.4

Axioms, interpretation and properties

We can construct a base probability distribution (the probability distribution) associated with this example using the simple events as follows:

Section 2.2

Examples:


P(E1) = 0.2

P(E2) = 0.1

P(E3) = 0.4

P(E4) = 0.3


Simple Event E1 E2 E3 E4P(E) 0.2 0.1 0.4 0.3Or

Simple Event O A B ABP(E) 0.2 0.1 0.4 0.3Or

We can use this probability distribution to find the probability of any event:

Section 2.2

Examples:



Simple Event O A B ABP(E) 0.2 0.1 0.4 0.3

Examples:

2) Studying the chance of observing the faces of one fair die when rolled:

Experiment:


Rolling a dieGoal: Observe faces of one die

{1, 2, 3, 4, 5, 6}

Events (subsets of S): (64 of them)


Probability distribution: This is a special case where each of the simple events are equally likely.

Examples:

2) Studying the chance of observing the faces of one fair die when rolled:


Simple Event 1 2 3 4 5 6P(E) 1/6 1/6 1/6 1/6 1/6 1/6

For any event A, P(A) in this case is given by:


Some other properties:1) For any event A:

P(A)+P(A’) = P(S) = 1 => P(A) = 1 – P(A’)

A’A

S

Note that and

Now, Axioms 2 and 3 will help us show this rule.


Some other properties:2) The probability of the empty set is:

Note that , and

Now, property (1) will help us show this property.


Some other properties:3) For any events A and B where :

B

S

A


Some other properties:4) For any event A:

Note that and use property (3)


Some other properties:5) For any events A and B:

A

S

B


Some other properties:Example:3) Forecasting the weather for each of the next three days

on the Palouse:

Experiment:


Observing if weather is rainy (R) or not (N) in each of those days.

{NNN, RNN, NRN, NNR, RRN, RNR, NRR, RRR}

Goal: Interested in whether you should bring an umbrella or not.


Some other properties:Example:

Let A be the event that day 3 is rainy, P(A) = 0.4

Let B be the event that day two is rainy, P(B) = 0.7

Let C be the event days 2 and 3 are rainy, P(C) = 0.3

Find probability that either day 2 or 3 will be rainy.

3) Forecasting the weather for each of the next three days on the Palouse:

ANNR, RNR

S

BNRN,RRN

CNRR,RRR


Some other properties:

Let A be the event that day three is rainy and P(A) = 0.4

Let B be the event that day two is rainy and P(B) = 0.7

Let C be the only days 2 and 3 are rainy and P(C) = 0.3

Find probability that either day 2 or 3 will be rainy.

= 0.4 + 0.7 – 0.3 = 0.8

Example:3) Forecasting the weather for each of the next three days

on the Palouse:







Find probability that day 3 and not day 2 is rainy.







Find probability that day 3 is not rainy.

sample space and events

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