Discrete Probability Distributions

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

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

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

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

Start with a frequency distribution 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.

Answer Probability Questions 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?

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 die Total of rolling two dice

Tossing coin and counting Heads One Coin Four Coins

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 • Business decisions

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

Discrete Probability Distributions