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. ). - PowerPoint PPT PresentationTRANSCRIPT
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%
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•
A specific simple example
# 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?
A specific simple example
# 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
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
# 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
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
Practice Calculations
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
• The standard deviation is
Practice Calculations
One CoinHow many heads Probability
0 1 / 2
1 1 / 2
Total 1
Statistics• The mean is
• The variance is
• The standard deviation is
Practice Calculations
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
• 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• 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
Expected Value Problems
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.”
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
Expected Value Problems
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
The Discrete Probability Distr.Outcome Net Value Probability
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
Expected Value Problems
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?
The Discrete Probability Distr.Outcome Net Value Probability
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