Randomness, Uncertainty, & Probability

Probability• The formal study of the laws of chance• Examples of probability statements are

everywhere:– There is a 60% chance of rain today– The chance of me winning the lottery is 1 in a

million– There is a 50-50 chance of observing a head when

a fair coin is tossed– There is a 1 in 6 chance in rolling a 6 with a single

die.

Randomness• Unpredictability– Cannot predict with any real certainty

Uncertainty• Outcomes are typically known• Which outcome will occur can be predicted w/

some certainty, but not 100%– I think I can, Maybe I can, I should be able to…

Probability• Outcomes are typically known• Which outcome will occur can be predicted w/

some certainty– Measure used to quantify the amount of “doubt”– Replaces the “I think; Maybe” statements with a

specific value of certainty.• I am 99% sure I can make the light• I am 0.60 sure I can hit the ball• I am 0.27 sure I will pass this class

Probability• Theoretically takes place in a Sample Space– All of the possible outcomes listed in a set of

brackets– Ex: When a child is born, the sample space would

be { B, G} • In a two child family, the sample space would be { BB,

BG, GB, GG}

• Can use matrix or tree graph to explore more complex outcomes

Sample Space (Graphically Displayed)

BB GB

BG GG

First Child Outcomes: Boy (B) Girl (G)

Seco

nd C

hild

Out

com

es:

G

irl (G

)

B

oy (B

)

Sample Space (Graphically Displayed)

Possible Outcomes

Boy (B)

Girl (G)

Boy (B)

Boy (B)

Girl (G)

Girl (G)

Sample SpaceBB

BG

GB

GG

If the 1st child is

And the 2nd child is

Probability• Theoretically each outcome of a Sample Space

is an Event– If only interested in the outcome BG, then, BG

would be the event of interest

Classical Probability

• We will denote the probability of an event E as P(E)– Roll a die: P(2), P(5)– Deck of cards: P(Ace), P(Spade)– Roll 2 dice: P(2,6)

• The previous formula will then be denoted as:

• Where n(E) is the number of events E• And n(S) is the total number of events in the sample

space

)(

)()(

Sn

EnEP

Classical Probability• Ex: If a two-child family is selected at random, what is

the probability of there being one boy and one girl with the girl born 1st?– Simple Event = GB = 1 Sample Space = {BB, BG, GB, GG} = 4– So: P(GB) = 1 = 0.25

4– The probability of there being one boy and 1 girl with the girl

born 1st in a two-child family is 0.25 or 25 %.

Classical Probability

Rolling Two Dice• What is P(2,5)?

• What is the P(Sum of Dice is a 7)?

Sample Space

03.036

1

)(

)5,2()(

Sn

nEP

17.036

6

)(

)7 is Sum()(

Sn

nEP

There is a .17 or 17% chance of rolling a pair of dice and the sum of the two being 7.

Empirical Probability

• AKA: Relative Frequency• The probability of an event occurring is the proportion

of times the event occurs over a given number of trials– Trials must be repeated exactly (norm, control)

• So: P(E) = frequency for the class number of trials

n

fEP )(

Empirical Probability• Ex: For the 1st 43 presidents of the US, 26 were lawyers.

What is the probability of randomly selecting a lawyer from the entire group of 43?

– (EA) = the event of a president being a lawyer – So: P(E) = f 26 = 0.61

n 43– The probability of randomly selecting a lawyer from the past

elected presidents is 0.61 or 61 %.

Empirical Probability• Ex: During the flu season, a health clinic observed that on one

day, 12 of 60 students examined had strep throat, whereas one week later 18 of 75 students examined had strep throat. What is the relative frequency for each given day?– (EA) = Relative Frequency Day 1

– (EB) = Relative Frequency Day 2

– So: P(EA) = 12 = 0.20 60

– P(EB) = 18 = 0.24 75

– If the data were collected over multiple day, the clinic could average the relative frequencies (generalize) to make one general statement:• During flu season, a student who is examined will have strep

throat 0.22 or 22% of the time.

Law of Large Numbers• The empirical probability for an event will

change from trial to trial– When repeated a large number of times, the

relative frequency approaches the classical probability for the event.

Subjective Probability• Probability measure of belief– Depends on life experiences of the subject.

• Sample must be explicitly defined

• Cannot be generalized outside the description of the sample

• EX: What are the chances that I will have an umbrella when it rains?

• Ex: What is the probability patients in cancer remission believe they will live beyond

the next 15 years?

General Probability Rules• Law 1: If the probability of an event is 1.00 or 100%,

then the event MUST occur.• Law 2: If the probability of an event is 0.00 or 0%,

then the event MUST NEVER occur. • Law 3: The probability of any event must assume a

value between 0.00 and 1.00.• Law 4: The sum of the probabilities of all the simple

events in a sample space must equal 1.– If there are 8 simple events, each event has a 1/8

chance of occurring. • If we sum these probabilities, we have 8 X 1/8 = 1.00

General Probability Rules• The closer the probability to 1.00, the more likely it is to

occur.• The closer the probability to 0.00, the less likely it is to

occur. • Compound Event– An event that is defined by combining two or more events.

• Let: A = students owning a laptop B = students owning an iPhone

C = students owning a laptop & an iPhone

To Discuss the event C, the researcher looks at the commonalities of both A and B.