Probability

CSC 172

SPRING 2002

LECTURE 12

Probability Space

Set of points, each with an attached probability (non-negative, real number) such that the sum of the probabilities is 1

Simplification

The number of points if finite, n

Each point has equal probability 1/n

Example: A deck of playing cards has 52 “points”

Experiments

An experiment is the selection of a point in probability space

With the “equally likely” simplification, all points have the same chance of being chosen

Example: Pick a card, any card . . .

Events

An event is an set of points in a probability spaceProb(E), the probability of an event E is the fraction

of the points in E

Example: E4 = a 4 is dealt.

Prob(E4) = 4/52 = 1/13

Example: Eh = a heart is dealt.

Prob(Eh) = 13/52 = 1/4

In general

Computing the probability of an event E involves two counts

1. The entire probability space

2. The number of points in E

Conditional Probability

Given that the outcome of an experiment is known to be some event E, what is the probability that the outcome is also some other event F?

Known as the conditional probability of F given E, or Prob(F|E)

== the fraction of the points in E that are also in F

Example: Pick a card, any card

Probability space contains 52 points

E = “a number card is selected”

= 36 points (deuce to 10, 4 suits)

F = “the card is less than 7”

= 20 points (deuce to 6, 4 suits)

Of the 36 points in E, 20 are also in F

Prob(F|E) = 20/36 = 5/9

Independent Events

F is independent of E if Prob(F|E) = Prob(F)

It makes on difference whether E occurs

Example:

E = “suit is hearts”

F = “rank is 5”

Prob(F|E) = 1/13

Prob(F) = 4/52 = 1/13

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Example

E = “suit is hearts”

F = “rank is 5”

Prob(F|E) = 1/13

Prob(F) = 4/52 = 1/13

Prob(E) = 13/52 = ¼

Prob(E|F) = 1/4

Complementary Events

If E is an event ~E is the event “E does not occur”

a.k.a E

Prob(~E) = 1 – Prob(E)

If F is independent of E, then F is independent of ~E

Expected ValueF is some function of points in a probability spaceSometimes “payoff”EV(f) = the average over points p of f(p)

Example: Would you take the following bet?You bet $8, and draw a card from a 52 card deckThe house pays you the value of the card in dollars2=$2, 3=$3, … 9=$9, 10=$10,J=$10,Q=$10,K=$10,A=$11 (“Blackjack values”)

Example Bet

Space = 52 cards

f = Blackjack value

EV(f) = (4*2+4*3+…+4*9+16*10+4*11)/52

EV(f)= 7.31

A tax on stupidity?NYS Lotto

For $1 make 2 Picks of 6 numbers 1-59

How much does the jackpot have to be before it makes sense to play?

Space = 59*58*57*56*55*54

= 32,441,381,280

EV(NYSlotto) = payoff/space

So, when jackpot is >= $17B, EV(NYSL) >= .50

When the jackpot is $10M, EV(NYSL) about 0.0001

Well, not exactlyIn NYS Lotto:

First Prize 75.00%

Second Prize 7.25%

Third Prize 5.50%

Fourth Prize 6.25%

Fifth Prize 6.00%

Also, you may have to pay 50% in taxes

don’t forget lawyers, bodyguards, etc.

So, now, what does the jackpot have to be? about $45B

Other considerations

Entertainment value

Charity aspect

- “good for education”

Social disparity issues

Gambling addiction

Counting trick

Describe the desired objects by a sequence of steps in which a choice is made from some number of options.

Example: The California lottery has announced that every ticket with 3 out of 6 numbers correct will win a “chance to win a” trip to Mexico. What are the odds of selecting exactly 3 numbers (1-53) correct out of 6?

Basic techinique

Count the total number of possibilities

Count the number of successful possibilites

Take the quotient

Basic techniqueTotal = (53,6) = 53!/47! = 16,529,385,600

Select 3 out of 6

a) Select 3 of the positions for the right guesses

b) Select 3 of the 6 numbers drawn

c) Select 3 out of 47 wrong numbers

a = 6 choose 3= 20, b =P(6,3) = 120, c = P(47,3) = 97,290

20 * 120 * 97290 = 233,496,000

Prob(“winning”) = 233,496,000/16,529,385,600

= 1.4%