Slides: Randomness in Data

RANDOMNESS IN DATA

ST101 – DR. ARIC LABARR

PROBABILITY (AND RISK)

RANDOMNESS IN DATA

CHANCE

Random – an outcome is random if we know the particular outcomes that something could have but are unsure of which of those outcomes is about to happen.

People throughout history have tried to measure patterns in randomness and answer the question, “what would happen if we did this many times?”

CHANCE

Try flipping a coin.

Each flip is completely random 🡪 you are unsure of the specific outcome.

If your coin is fair (evenly weighted), then in many flips you should get approximately 50% heads and 50% tails.

PROBABILITY

The probability that an event happens is a numerical measure of the likelihood of that event’s occurrence.

0

0.5

1

Likelihood of Occurrence

Probability

The event is

very unlikely 

to happen.

The event is

equally likely 

to happen as

unlikely to.

The event is

very likely

to happen.

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PROBABILITY

The probability that an event happens is a numerical measure of the likelihood of that event’s occurrence.

Probabilities are numbers between 0 and 1. 

Percentages are numbers between 0 and 100.

Sample space: the collection of all possible outcomes in a random process.

Sum of all probabilities for an experiment must sum to 1.

6

EVENTS

An event is a collection of one or more outcomes from a process whose result cannot be predicted with certainty.

The probability of an event A is denoted, P(A).

Examples: 

When flipping a fair dice, what is the probability of it landing on heads?

When rolling a fair dice, what is the probability of rolling a 6?

MULTI-STEP RANDOM PROCESSES

TREE DIAGRAMS

Multi-step random processes can be visualized easily with tree diagrams.

For example, you have a 2-step random process where you flip a coin twice:

Flip Coin

Heads – Flip

Tails

Tails – Flip

Tails

Heads

Heads

Probabilities of an event occurring must be between 0 and 1. 

The sum of the probabilities of all events in an experiment must equal 1.

There are three typical methods for assigning probabilities to events:

Classical Method

Relative Frequency Method

Subjective Method

ASSIGNING PROBABILITIES

CLASSICAL METHOD

RELATIVE FREQUENCY METHOD

The relative frequency method of assigning probabilities assigns probabilities based on experimentation or historical data.

For example, you don’t believe that I have a fairly weighted dice so you ask me to roll it 100 times and get the following:

Value of the Roll

Frequency

Experimental Probability

1

10

0.10

2

25

0.25

3

42

0.42

4

7

0.07

5

10

0.10

6

6

0.06

Circumstances might change rapidly in the events you are trying to build probabilities for, so things shouldn’t be based solely on historical data.

Use both a combination of historical data values as well experience and intuition about how likely an event will be to occur.

Best probability estimates are typically a combination of subjective and classical/relative frequency methods.

SUBJECTIVE METHOD

An outcome is random if we know the particular outcomes that something could have but are unsure of which of those outcomes is about to happen.

The probability that an event happens is a numerical measure of the likelihood of that event’s occurrence.

Classical Method

Relative Frequency Method

Subjective Method

An event is a collection of one or more outcomes from a process whose result cannot be predicted with certainty.

SUMMARY

LAW OF LARGE NUMBERS

RANDOMNESS IN DATA

CHANCE

Try flipping a coin.

Each flip is completely random 🡪 you are unsure of the specific outcome.

If your coin is fair (evenly weighted), then in many flips you should get approximately 50% heads and 50% tails.

Chance behavior is unpredictable in the short run, but predictable in the long run.

EXAMPLE – TOSS A COIN

Toss a coin 500 times and record the proportion of heads as you go.

Early on, the proportion of heads can vary drastically.

In the long run, it goes to what we expect.

If we flip coin enough times, the overall proportion of times it lands on heads (or tails) gets closer to 50%.

Reasonable to assume that it is a fair coin – half of the time it lands on heads.

The law of large numbers states that as the number of independent trials increases, in the long run the proportion for a certain event gets closer and closer to a single value (the probability of the event).

LAW OF LARGE NUMBERS

LAW OF LARGE NUMBERS

The law of large numbers states that as the number of independent trials increases, in the long run the proportion for a certain event gets closer and closer to a single value (the probability of the event).

MYTH OF SHORT RUN PREDICTABILITY

Chance behavior is unpredictable in the short run, but predictable in the long run.

This is counter-intuitive to most people.

Example – Which of the following outcomes of flipping a fair coin 4 times is more probable?

H, T, T, H

H, H, H, H

MYTH OF SHORT RUN PREDICTABILITY

Chance behavior is unpredictable in the short run, but predictable in the long run.

This is counter-intuitive to most people.

Example – Which of the following outcomes of flipping a fair coin 4 times is more probable?

H, T, T, H

H, H, H, H

SAME!

MYTH OF SHORT RUN PREDICTABILITY

Heads – Flip

Heads – Flip

Tails – Flip

Tails – Flip

Tails – Flip

Heads – Flip

Heads – Flip

Heads

Tails

Heads

Tails

Heads

Tails

Heads

Tails

Flip Coin

Tails – Flip

MYTH OF SHORT RUN PREDICTABILITY

Heads – Flip

Heads – Flip

Tails – Flip

Tails – Flip

Tails – Flip

Heads – Flip

Heads – Flip

Heads

Tails

Heads

Tails

Heads

Tails

Heads

Tails

Flip Coin

Tails – Flip

MYTH OF SHORT RUN PREDICTABILITY

Heads – Flip

Heads – Flip

Tails – Flip

Tails – Flip

Tails – Flip

Heads – Flip

Heads – Flip

Heads

Tails

Heads

Tails

Heads

Tails

Heads

Tails

Flip Coin

Tails – Flip

MYTH OF THE “HOT HAND”

Chance behavior is unpredictable in the short run, but predictable in the long run.

This is counter-intuitive to most people.

Example – A player in basketball makes 4 straight shots as compared to makes 1, misses 2, then makes 1. Same idea as the coins!

H, T, T, H

H, H, H, H

MYTH OF THE “LAW OF AVERAGES”

Chance behavior is unpredictable in the short run, but predictable in the long run.

This is counter-intuitive to most people.

Example – A couple has four kids that are all boys. They think their 5th kid must “be due” to have a girl. Same idea as the coins! Still 50/50.

MYTH OF THE “LAW OF AVERAGES”

Chance behavior is unpredictable in the short run, but predictable in the long run.

This is counter-intuitive to most people.

The “law of averages” doesn’t exist 🡪 things even out in the long run, but chances of outcomes do not change.

Chance behavior is unpredictable in the short run, but predictable in the long run.

The law of large numbers states that as the number of independent trials increases, in the long run the proportion for a certain event gets closer and closer to a single value (the probability of the event).

The “law of averages” doesn’t exist 🡪 things even out in the long run, but chances of outcomes do not change.

SUMMARY

BASIC PROBABILITY RULES

RANDOMNESS IN DATA

BASIC RELATIONSHIPS

You do not always know all of the sample point probabilities in an event.

However, there are some basic probability relationships that still can be used to calculate the probability of an event occurring:

Complement of an Event

Union of Two Events

Intersection of Two Events

Mutually Exclusive Events

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COMPLEMENT OF AN EVENT

Sample Space

Event A

UNION OF TWO EVENTS

Event A

Event B

Event A

Sample Space

UNION OF TWO EVENTS

Event A

Event B

Event A

Sample Space

INTERSECTION OF TWO EVENTS

Event B

Event A

Intersection

Sample Space

The addition law provides a way to compute the union of events A and B:

ADDITION LAW

MUTUALLY EXCLUSIVE EVENTS

Two events are mutually exclusive if the events have no sample points in common – do not intersect.

This also means that the events cannot both occur. If one event occurs, the other cannot.

Event B

Event A

Sample Space

The addition law provides a way to compute the union of events A and B:

ADDITION LAW – MUTUALLY EXCLUSIVE EVENTS

If two events are mutually exclusive

they do not intersect.

EXAMPLE

Weather

Spring

Summer

Fall

Winter

Total

Clear of Cloudy

626,986

799,443

519,487

312,036

2,257,952

Misty

288,096

250,679

302,510

155,573

996,858

Rain or Snow

3,507

11,007

19,616

3,739

37,869

Total

918,589

1,061,129

841,613

471,348

3,292,679

38

EXAMPLE

What is the probability a random customer uses the bike service in Fall and it was raining or snowing?

39

EXAMPLE

Weather

Spring

Summer

Fall

Winter

Total

Clear of Cloudy

626,986

799,443

519,487

312,036

2,257,952

Misty

288,096

250,679

302,510

155,573

996,858

Rain or Snow

3,507

11,007

19,616

3,739

37,869

Total

918,589

1,061,129

841,613

471,348

3,292,679

40

EXAMPLE

What is the probability a random customer uses the bike service in Fall and it was raining or snowing?

41

EXAMPLE

What is the probability a random customer uses the bike service in Fall or it was raining or snowing?

42

EXAMPLE

Weather

Spring

Summer

Fall

Winter

Total

Clear of Cloudy

626,986

799,443

519,487

312,036

2,257,952

Misty

288,096

250,679

302,510

155,573

996,858

Rain or Snow

3,507

11,007

19,616

3,739

37,869

Total

918,589

1,061,129

841,613

471,348

3,292,679

43

EXAMPLE

What is the probability a random customer uses the bike service in Fall or it was raining or snowing?

44

SUMMARY

CONDITIONAL PROBABILITIES

RANDOMNESS IN DATA

CONDITIONAL PROBABILITIES

The multiplication law provides a way to compute the probability of the intersection of two events as long as you know the conditional probabilities:

MULTIPLICATION LAW

OR

If the probability of an event A is not changed by the existence of event B, then the two events are called independent.

INDEPENDENT EVENTS

OR

The multiplication law provides a way to compute the probability of the intersection of two events as long as you know the conditional probabilities:

MULTIPLICATION LAW – INDEPENDENT EVENTS

ONLY IF EVENTS ARE INDEPENDENT

TREE DIAGRAMS

Tree diagrams can help calculate probabilities in a series of independent events.

For example, you have a 2-step random process where you flip a coin twice (independent flips):

Flip Coin

Heads – Flip

Tails

Tails – Flip

Tails

Heads

Heads

TREE DIAGRAMS

Tree diagrams can help calculate probabilities in a series of independent events.

For example, you have a 2-step random process where you flip a coin twice (independent flips):

Flip Coin

Heads – Flip

Tails

Tails – Flip

Tails

Heads

TREE DIAGRAMS

Tree diagrams can help calculate probabilities in a series of independent events.

For example, you have a 2-step random process where you flip a coin twice (independent flips):

Flip Coin

Heads – Flip

Tails – Flip

EXAMPLE

Weather

Spring

Summer

Fall

Winter

Total

Clear of Cloudy

626,986

799,443

519,487

312,036

2,257,952

Misty

288,096

250,679

302,510

155,573

996,858

Rain or Snow

3,507

11,007

19,616

3,739

37,869

Total

918,589

1,061,129

841,613

471,348

3,292,679

54

EXAMPLE

What is the probability a random customer uses the bike service in on a misty day given it is winter?

55

EXAMPLE

Weather

Spring

Summer

Fall

Winter

Total

Clear of Cloudy

626,986

799,443

519,487

312,036

2,257,952

Misty

288,096

250,679

302,510

155,573

996,858

Rain or Snow

3,507

11,007

19,616

3,739

37,869

Total

918,589

1,061,129

841,613

471,348

3,292,679

56

EXAMPLE

What is the probability a random customer uses the bike service in on a misty day given it is winter?

57

EXAMPLE

What is the probability a random customer uses the bike service in on a misty day given it is winter?

OR

58

The probability of an event given that another event has occurred is called a conditional (or joint) probability.

The multiplication law provides a way to compute the probability of the intersection of two events as long as you know the conditional probabilities.

If the probability of an event A is not changed by the existence of event B, then the two events are called independent.

SUMMARY