Probability and Long-Term Expectations

Goals

Understand the concept of probabilityGrasp the idea of long-term relative frequency as probabilityLearn some simple probability rulesUnderstand how hard it is to win lotteries like Euro Millions

Probability

Two distinct concepts:Relative frequency interpretationPersonal probability interpretation

Relative Frequency

The probability of an outcome is defined as the proportion (percentage) of times the outcome occurs over the long run.

Boy frequency in 25 births

0.4500.5000.5500.6000.6500.7000.7500.8000.8500.9000.9501.000

1 3 5 7 9 11 13 15 17 19 21 23 25

Boy frequency in 200 births

0.450

0.500

0.550

0.600

0.650

0.700

0.750

0.800

1 18 35 52 69 86 103 120 137 154 171 188

Boy frequency in 5,000 births

0.4500.5000.5500.6000.6500.7000.7500.8001

417

833

1249

1665

2081

2497

2913

3329

3745

4161

4577

4993

births

bo

y fr

eq

Two Ways to DetermineRelative Frequency

Make physical assumptions coins, cards, dice, lottery numbers,

etc.

Make repeated observations births, cancer, weather

Personal Probability

Personal probability is the degree to which an individual believes some event will happenUseful for predicting the likelihood of events that aren’t repeatable -- accurately or not

Which kind of probability?

A lottery ticket will be a winner.You will get an B.A random student will get a B.The Lisbon-Madrid flight will leave on time.Portugal will win the next CopaSomeone in this class will live to be at least 90.

Probability Definitions

The probability of something occurring can never be less than zero or more than one.If two outcomes can’t happen at the same time, they are mutually exclusive.If two events don’t influence each other, the events are independent of each other.

Probability Rule 1

If there are only two possible outcomes, their probabilities must add to 1.

Examples:Heads is 0.5, tails is...?Boy birth is 0.51, girl birth is...?Card a club is 0.25, not a club is…?Plane on time is 0.80, late is…?

Probability Rule 2

With mutually exclusive outcomes, the probability of one or the other happening is the sum of their individual probabilities.

Examples: age at first birth (.25 under 20, .33 for

20-24) heart attack (0.30) or cancer (0.23)

Probability Rule 3

If two events are independent, the probability they both happen is found by multiplying the individual probabilities.

Examples: kids’ genders Student smokers

Independent probabilities

Remember that dice, lottery machines, etc., don’t remember what they have done in the past.Each roll or draw or whatever is independent, so the probability DOESN’T change

“Ask Marilyn” problem

A woman and a man (unrelated) each have two children. At least one of the woman’s children is a boy, and the man’s older child is a boy.

Do the chances that the woman has two boys equal the chances that the man has two boys?

Answer

Woman:boy -- girlgirl -- boyboy -- boy

Man:boy -- girlboy -- boy

Probability Rule 4

If the ways one event can occur are a subset of the ways another can occur, then the probability of the first event occurring cannot be higher than the second.

Example: death by accident or in a car crash

Class Survey

Which is more likely to occur in the next 10 years?:A nuclear war

orUse of nuclear weapons in the Middle East sparked by a terrorist attack

Class Survey

Which is more likely to occur in the next 10 years?:A nuclear war (22%)

orUse of nuclear weapons in the Middle East sparked by a terrorist attack (78%)

Long-Term Probabilities

If probability of an outcome is p, and the number of trials is n:Chance of it occurring in n trials:

1 - (1-p)n

Chance of it occurring on the nth trial:

p * (1-p)n-1

Some Long-Term Probabilities

Chance of rolling a 6 is 1/6

Rolling a 6

p = 1/6 = 0.167Chance of rolling a 6 in 5 rolls:

1-(1- ,167)5 = 1- (,833)5 = ,60Chance of rolling a 6 on the 5th roll:

,167 * (,833)4 = ,08

Some Long-Term Probabilities

Chance of rolling a 6 is 1/6Chance of dealing the ace of spades is 1/52

Dealing the Ace of Spades

p = 1/52 = 0.019Chance of dealing it in 20 tries:

1-(1-,019)20 = 1-(,981)20 = ,32Chance of dealing it as the 20th card:

,019 * (,981)19 = ,013

Some Long-Term Probabilities

Chance of rolling a 6 is 1/6Chance of dealing the ace of spades is 1/52Risk of heterosexual HIV transmission in unprotected sex is about 1/1000.

HIV transmission

p = 1/1000 = 0.001Chance of transmission in 4 encounters:

1-(1 - ,001)4 = 1-(,999)4 = ,004Chance in 10 encounters:

(1 - ,001)10 = (,999)10 = ,009Chance in 50 encounters:

(1 - ,001)50 = (,999)50 = ,049

Some Long-Term Probabilities

Chance of rolling a 6 is 1/6Chance of dealing the ace of spades is 1/52Risk of HIV transmission from female to male in unprotected sex is about 1/400.Risk of space shuttle accident is 2/119.

Space Shuttle Accident

p = 2/119 = 0.0168Chance of accident in next 25 launches:

1-(1- ,0168)25 = 1-(,982)25

=.35

Euro Millions lottery

Odds of winning: 1 / 76.275.360Lay tickets end to end: About 6.000 kmLisbon>Madrid>Paris About 1.500 km

Remember

The Lottery is a tax on people who can’t do math.