Probability and Long- Term Expectations

Goals Understand the concept of probability Grasp the idea of long-term relative frequency as probability Learn some simple probability rules Understand how hard it is to win lotteries like Euro Millions

Probability Two distinct concepts: Relative frequency interpretation Personal 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

Boy frequency in 200 births

Boy frequency in 5,000 births

Two Ways to Determine Relative 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 happen Useful 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 Copa Someone 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 ) 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 -- girl girl -- boy boy -- boy Man: boy -- girl boy -- 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 or Use 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%) or Use 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 = Chance of rolling a 6 in 5 rolls: 1-(1-,167) 5 = 1- (,833) 5 =,60 Chance of rolling a 6 on the 5th roll:,167 * (,833) 4 =,08

Some Long-Term Probabilities Chance of rolling a 6 is 1/6 Chance of dealing the ace of spades is 1/52

Dealing the Ace of Spades p = 1/52 = Chance of dealing it in 20 tries: 1-(1-,019) 20 = 1-(,981) 20 =,32 Chance of dealing it as the 20th card:,019 * (,981) 19 =,013

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

HIV transmission p = 1/1000 = Chance of transmission in 4 encounters: 1-(1 -,001) 4 = 1-(,999) 4 =,004 Chance in 10 encounters: (1 -,001) 10 = (,999) 10 =,009 Chance in 50 encounters: (1 -,001) 50 = (,999) 50 =,049

Some Long-Term Probabilities Chance of rolling a 6 is 1/6 Chance of dealing the ace of spades is 1/52 Risk 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 = Chance of accident in next 25 launches: 1-(1-,0168) 25 = 1-(,982) 25 =.35

Euro Millions lottery Odds of winning: 1 / Lay tickets end to end: About km Lisbon>Madrid>Paris About km

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