Discrete Probability Distributions

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Discrete Probability Distributions. To accompany Hawkes lesson 5.1 Original content by D.R.S. Examples of Probability Distributions. Rolling a single die. Total of rolling two dice. ( Note that it’s a two-column chart but we had to typeset it this way to fit it onto the slide. ).

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Discrete Probability Distributions

To accompany Hawkes lesson 5.1

Original content by D.R.S.

Examples of Probability Distributions

Rolling a single die

Total of rolling two dice

(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
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

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

• 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
Discrete is our focus for now

Discrete

Continuous

Will talk about continuous probability distributions in future chapters.

• 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

General layout

A specific made-up example

Include a Relative Frequency column

General layout

A specific simple example

You can drop the count column

General layout

A specific simple example

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.

What is the probability …

A specific simple example

• …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?

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 die

Total of rolling two dice

Tossing coin and counting Heads

How did we get this?

Four Coins

• 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 Coins

Probability

6/16

4/16

1/16

0 1 2 3 4 heads

Shape of the distribution

Histogram, for example

Distribution shapes matter!

This one is a bell-shaped distribution

Rolling a single die: its graph is a uniform distribution

Other distribution shapes can happen, too

Probability

6/16

3/16

1/16

0 1 2 3 4 heads

Remember the Structure

Required features

Example of a

Discrete Probability Distribution

• 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.
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 die

Statistics

The mean is

The variance is

The standard deviation is

Practice Calculations

Statistics

Total of rolling two dice

• The mean is
• The variance is
• The standard deviation is
Practice Calculations

One Coin

Statistics

The mean is

The variance is

The standard deviation is

Practice Calculations

Statistics

Four Coins

• The mean is
• The variance is
• The standard deviation 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
Expected Value Problems

The Situation

The Discrete Probability Distr.

• 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?

Expected Value Problems

Statistics

The Discrete Probability Distr.

• 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.”
Expected Value Problems

Another way to do it

The Discrete Probability Distr.

• 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
Expected Value Problems

The Situation

The Discrete Probability Distr.

• We’re the insurance company.
• We sell an auto policy for \$500 for 6 months coverage on a \$20,000 car.
• The deductible is \$200

What is the “expected value” – that is, profit – to us, the insurance company?

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