Discrete Probability Distributions

Discrete Probability Distributions

To accompany Hawkes lesson 5.1Original content by D.R.S.

Examples of Probability Distributions

Rolling a single die Total of rolling two diceValue Prob. Value Prob.

2 1/36 8 5/36

3 2/36 9 4/36

4 3/36 10 3/36

5 4/36 11 2/36

6 5/36 12 1/36

7 6/36 Total 1

Value Probability

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

Total 1(Note that it’s a two-column chart but we had to typeset it this way to fit it onto the slide.)

Example of a Probability Distributionhttp://en.wikipedia.org/wiki/Poker_probability

Draw this 5-card poker hand Probability

Royal Flush 0.000154%

Straight Flush (not including Royal Flush) 0.00139%

Four of a Kind 0.0240%

Full House 0.144%

Flush (not including Royal Flush or Straight Flush) 0.197%

Straight (not including Royal Flush or Straight Flush) 0.392%

Three of a Kind 2.11%

Two Pair 4.75%

One Pair 42.3%

Something that’s not special at all 50.1%

Total (inexact, due to rounding) 100%

http://en.wikipedia.org/wiki/Poker_probability

Exact fractions avoid rounding errors (but is it useful to readers?)

Draw this 5-card poker hand Probability

Royal Flush 4 / 2,598,960

Straight Flush (not including Royal Flush) 36 / 2,598,960

Four of a Kind 624 / 2,598,960

Full House 3,744 / 2,598,960

Flush (not including Royal Flush or Straight Flush) 5,108 / 2,598,960

Straight (not including Royal Flush or Straight Flush) 10,200 / 2,598,960

Three of a Kind 54,912 / 2,598,960

Two Pair 123,552 / 2,598,960

One Pair 1,098,240 / 2,598,960

Something that’s not special at all 1,302,540 / 2,598,960

Total (exact, precise, beautiful fractions) 2,598,600 / 2,598,600

Example of a probability distribution“How effective is Treatment X?”

Outcome ProbabilityThe patient is cured. 85%The patient’s condition improves. 10%There is no apparent effect. 4%The patient’s condition deteriorates. 1%

A Random Variable

• The value of “x” is determined by chance• Or “could be” determined by chance• As far as we know, it’s “random”, “by chance”

• The important thing: it’s some value we get in a single trial of a probability experiment

• It’s what we’re measuring

Discrete vs. Continuous

Discrete• A countable number of

values

• “Red”, “Yellow”, “Green”

• 2 of diamonds, 2 of hearts, … etc.

• 1, 2, 3, 4, 5, 6 rolled on a die

Continuous• All real numbers in some

interval

• An age between 10 and 80 (10.000000 and 80.000000)

• A dollar amount

• A height or weight

Discrete is our focus for now

Discrete• A countable number of

values (outcomes)• “Red”, “Yellow”, “Green”• “Improved”, “Worsened”• 2 of diamonds, 2 of hearts,

… etc.• What poker hand you draw.• 1, 2, 3, 4, 5, 6 rolled on a die• Total dots in rolling two dice

Continuous• Will talk about continuous

probability distributions in future chapters.

Start with a frequency distribution

General layout•

A specific made-up example

How many children live here?

Number of households

0 50

1 100

2 150

3 80

4 40

5 20

6 or more 10

Total responses 450

Outcome Count of occurrences

Include a Relative Frequency column

General layout•

A specific simple example

# of children

Number of households

Relative Frequency

0 50 0.108

1 110 0.239

2 150 0.326

3 80 0.174

4 40 0.087

5 20 0.043

6+ 10 0.022

Total 460 1.000

Outcome Count of occur-rences

RelativeFrequency=count ÷ total

You can drop the count column

General layout•


# of children Relative Frequency

0 0.108

1 0.239

2 0.326

3 0.174

4 0.087

5 0.043

6+ 0.022

Total 1.000

Outcome RelativeFrequency=count ÷ total

Sum MUST BE EXACTLY 1 !!!

• In every Probability Distribution, the total of the probabilities must always, every time, without exception, be exactly 1.00000000000.– In some cases, it might be off a hair because of

rounding, like 0.999 for example.– If you can maintain exact fractions, this rounding

problem won’t happen.

Answer Probability Questions

What is the probability …• …that a randomly selected

household has exactly 3 children?

• …that a randomly selected household has children?

• … that a randomly selected household has fewer than 3 children?

• … no more than 3 children?



0 0.108

1 0.239

2 0.326

3 0.174

4 0.087

5 0.043

6+ 0.022

Total 1.000

Answer Probability Questions

Referring to the Poker probabilities table• “What is the probability of drawing a Four of a

Kind hand?”• “What is the probability of drawing a Three of a

Kind or better?”• “What is the probability of drawing something

worse than Three of a Kind?”• “What is the probability of a One Pair hand

twice in a row? (after replace & reshuffle?)”

Theoretical Probabilities

Rolling one dieValue Probability

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

Total 1

Total of rolling two diceValue Prob. Value Prob.

2 1/36 8 5/36

3 2/36 9 4/36

4 3/36 10 3/36

5 4/36 11 2/36

6 5/36 12 1/36

7 6/36 Total 1

Tossing coin and counting Heads

One CoinHow many heads Probability

0 1 / 2

1 1 / 2

Total 1

Four CoinsHow many heads Probability

0 1/16

1 4/16

2 6/16

3 4/16

4 1/16

Total 1

Tossing coin and counting Heads

How did we get this? Four CoinsHow many heads Probability

0 1/16

1 3/16

2 6/16

3 3/16

4 1/16

Total 1

• Could try to list the entire sample space: TTTT, TTTH, TTHT, TTHH, THTT, etc.

• Could use a tree diagram to get the sample space.

• Could use nCr combinations.

• We will formally study The Binomial Distribution soon.

Graphical Representation

Histogram, for example Four CoinsHow many heads Probability

0 1/16

1 4/16

2 6/16

3 4/16

4 1/16

Total 1

6/16

4/16

1/16

0 1 2 3 4 heads

Probability

Shape of the distribution

Histogram, for example Distribution shapes matter!

6/16

3/16

1/16

0 1 2 3 4 heads

Probability• This one is a bell-shaped

distribution

• Rolling a single die: its graph is a uniform distribution

• Other distribution shapes can happen, too

Remember the Structure

Required features• The left column lists the

sample space outcomes.• The right column has the

probability of each of the outcomes.

• The probabilities in the right column must sum to exactly 1.0000000000000000000.

Example of a Discrete Probability Distribution


0 0.108

1 0.239

2 0.326

3 0.174

4 0.087

5 0.043

6+ 0.022

Total 1.000

The Formulas

• MEAN:

• VARIANCE:

• STANDARD DEVIATION:

TI-84 Calculations

• Put the outcomes into a TI-84 List (we’ll use L1)

• Put the corresponding probabilities into another TI-84 List (we’ll use L2)

• 1-Var Stats L1, L2

• You can type fractions into the lists, too!

•

Practice Calculations

Rolling one dieValue Probability

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

Total 1

Statistics• The mean is

• The variance is

• The standard deviation is


Statistics Total of rolling two diceValue Prob. Value Prob.

2 1/36 8 5/36

3 2/36 9 4/36

4 3/36 10 3/36

5 4/36 11 2/36

6 5/36 12 1/36

7 6/36 Total 1

• The mean is

• The variance is



One CoinHow many heads Probability

0 1 / 2

1 1 / 2

Total 1

Statistics• The mean is

• The variance is



Statistics Four CoinsHow many heads Probability

0 1/16

1 4/16

2 6/16

3 4/16

4 1/16

Total 1

• The mean is

• The variance is


Expected Value

• Probability Distribution with THREE columns– Event– Probability of the event– Value of the event (sometimes same as the event)

• Examples:– Games of chance– Insurance payoffs– Business decisions

Expected Value Problems

The Situation• 1000 raffle tickets are sold• You pay $5 to buy a ticket• First prize is $2,000• Second prize is $1,000• Two third prizes, each $500• Three more get $100 each• The other ____ are losers.What is the “expected value” of your ticket?

The Discrete Probability Distr.Outcome Net Value Probability

Win first prize

$1,995 1/1000

Win second prize

$995 1/1000

Win third prize

$495 2/1000

Win fourth prize

$95 3/1000

Loser $ -5 993/1000

Total 1000/1000


Statistics• The mean of this probability

is $ - 0.70, a negative value.• This is also called “Expected

Value”.

• Interpretation: “On the average, I’m going to end up losing 70 cents by investing in this raffle ticket.”


Win first prize

$1,995 1/1000

Win second prize

$995 1/1000

Win third prize

$495 2/1000

Win fourth prize

$95 3/1000

Loser $ -5 993/1000

Total 1000/1000


Another way to do it• Use only the prize values.• The expected value is the

mean of the probability distribution which is $4.30

• Then at the end, subtract the $5 cost of a ticket, once.

• Result is the same, an expected value = $ -0.70


Win first prize

$2,000 1/1000

Win second prize

$1,000 1/1000

Win third prize

$500 2/1000

Win fourth prize

$100 3/1000

Loser $ 0 993/1000

Total 1000/1000


The Situation• We’re the insurance

company.• We sell an auto policy for

$500 for 6 months coverage on a $20,000 car.

• The deductible is $200What is the “expected value” – that is, profit – to us, the insurance company?


No claims filed _______An $800 fender bender

0.004

An $8,000 accident 0.002A wreck, it’s totaled 0.002

An Observation

• The mean of a probability distribution is really the same as the weighted mean we have seen.

• Recall that GPA is a classic instance of weighted mean– Grades are the values– Course credits are the weights

• Think about the raffle example– Prizes are the values– Probabilities of the prizes are the weights

Documents

Discrete Probability Distributions