Introduction to Probability and Risk ISP 121 Introduction to Probability and Risk

A Question With terrorism, homicides, and traffic accidents, is it safer to stay home and take a college course online rather than head downtown to class? We’ll come back to this later

Three Basic Forms Theoretical, or a priori probability – based on a model in which all outcomes are equally likely. Probability of a die landing on a 2 = 1/6. Empirical probability – base the probability on the results of observations or experiments. If it rains an average of 100 days a year, we might say the probability of rain on any one day is 100/365.

Three Basic Forms Subjective (personal) probability – use personal judgment or intuition. If you go to college today, you will be more successful in the future.

Possible Outcomes Suppose there are M possible outcomes for one process and N possible outcomes for a second process. The total number of possible outcomes for the two processes combined is M x N. How many outcomes are possible when you roll two dice?

Possible Outcomes Continued A restaurant menu offers two choices for an appetizer, five choices for a main course, and three choices for a dessert. How many different three-course meals? A college offers 12 natural science classes, 15 social science classes, 10 English classes, and 8 fine arts classes. How many choices? 14400

Theoretical Probability P(A) = (number of ways A can occur) / (total number of outcomes) Probability of a head landing in a coin toss: 1/2 Probability of rolling a 7 using two dice: 6/36 Probability that a family of 3 will have two boys and one girl: 3/8 (BBB,BBG,BGB,BGG,GBB, GBG, GGB, GGG)

Empirical Probability Probability based on observations or experiments Records indicate that a river has crested above flood level just four times in the past 2000 years. What is the empirical probability that the river will crest above flood level next year? 4/2000 = 1/500 = 0.002

Probability of an Event Not Occurring P(not A) = 1 - P(A) If the probability of rolling a 7 with two dice is 6/36, then the probability of not rolling a 7 with two dice is 30/36

Combining Probabilities - Independent Events Two events are independent if the outcome of one does not affect the outcome of the next The probability of A and B occurring together, P(A and B), = P(A) x P(B)

Combining Probabilities - Independent Events For example, suppose you toss three coins. What is the probability of getting three tails? 1/2 x 1/2 x 1/2 = 1/8 Find the probability that a 100-year flood will strike a city in two consecutive years 1 in 100 x 1 in 100 = 0.01 x 0.01 = 0.0001

Combining Probabilities - Independent Events You are playing craps in Vegas. You have had a string of bad luck. But you figure since your luck has been so bad, it has to balance out and turn good Bad assumption! Each event is independent of another and has nothing to do with previous run. Especially in the short run (as we will see in a few slides) This is called Gambler’s Fallacy Is this the same for playing Blackjack?

Either/Or Probabilities - Non-Overlapping Events If you ask what is the probability of either this happening or that happening, and the two events don’t overlap: P(A or B) = P(A) + P(B) Suppose you roll a single die. What is the probability of rolling either a 2 or a 3? P(2 or 3) = P(2) + P(3) = 1/6 + 1/6 = 2/6

Probability of At Least Once What is the probability of something happening at least once? P(at least one event A in n trials) = 1 - [P(not A in one trial)]n

Example What is the probability that a region will experience at least one 100-year flood during the next 100 years? Probability of a flood is 1/100. Probability of no flood is 99/100. P(at least one flood in 100 years) = 1 - 0.99100 = 0.634

Another Example You purchase 10 lottery tickets, for which the probability of winning some prize on a single ticket is 1 in 10. What is the probability that you will have at least one winning ticket? P(at least one winner in 10 tickets) = 1 - 0.910 = 0.651

Expected Value The probability of tossing a coin and landing tails is 0.5. But what if you toss it 5 times and you get HHHHH? The law of large numbers tells you that if you toss it 100 / 1000 / 1,000,000 times, you should get 0.5. But this may not be the case if you only toss it 5 times.

Expected Value Furthermore, what if you have multiple related events? What is the expected value from the set of events? Expected value = event 1 value x event 1 probability + event 2 value x event 2 probability + …

Example Suppose that $1 lottery tickets have the following probabilities: 1 in 5 win a free $1 ticket; 1 in 100 win $5; 1 in 100,000 to win $1000; and 1 in 10 million to win $1 million. What is the expected value of a lottery ticket?

Example - Solution Ticket purchase: value -$1, prob 1 Win free ticket: value $1, prob 1/5 Win $5: value $5, prob 1/100 Win $1000: prob 1/100,000 Win $1million: prob 1/10,000,000 -$1 x 1= -1; $1 x 1/5 = $0.20; $5 x 1/100 = $0.05; $1000 x 1/100,000 = $0.01; $1,000,000 x 1/10,000,000 = $0.10

Solution Continued Now sum all the products: -$1 + 0.20 + 0.05 + 0.01 + 0.10 = -$0.64 Thus, averaged over many tickets, you should expect to lose $0.64 for each lottery ticket that you buy. If you buy, say, 1000 tickets, you should lose $640.

Another Example – Expected Value Suppose an insurance company sells policies for $500 each. The company knows that about 10% will submit a claim that year and that claims average to $1500 each. How much can the company expect to make per customer?

Another Example – Expected Value Company makes $500 100% of the time (when a policy is sold) Company loses $1500 10% of the time $500 x 1.0 - $1500 x 0.1 = 500 – 150 = 350 Company gains $350 from each customer The company needs to have a lot of customers to ensure this works

Do You Take Risks? Are you safer in a small car or a sport utility vehicle? Are cars today safer than those 30 years ago? If you need to travel across country, are you safer flying or driving?

The Risk of Driving In 1966, there were 51,000 deaths related to driving, and people drove 9 x 1011 miles In 2000, there were 42,000 deaths related to driving, and people drove 2.75 x 1012 miles Was driving safer in 2000?

The Risk of Driving 51,000 deaths / 9 x 1011 miles = 5.7 x 10-8 deaths per mile 42,000 deaths / 2.75 x 1012 miles = 1.5 x 10-8 deaths per mile Driving has gotten safer! Why?

Driving vs. Flying Over the last 20 years, airline travel has averaged 100 deaths per year Airlines have averaged 7 billion miles in the air 100 deaths / 7 billion miles = 1.4 x 10-8 deaths per mile How does this compare to driving?

The Certainty Effect Suppose you are buying a new car. For an additional $200 you can add a device that will reduce your chances of death in a highway accident from 50% to 45%. Interested? What if the salesman told you it could reduce your chances of death from 5% to 0%. Interested now? Why?

The Certainty Effect Suppose you can purchase an extended warranty plan which covers some items completely but other items not at all Or you can purchase an extended warranty plan which covers all items at 30% coverage Which would you choose?

The Availability Heuristic Which do you think caused more deaths in the US in 2000, homicide or diabetes? Homicide: 6.0 deaths per 100,000 Diabetes: 24.6 deaths per 100,000

Which Has More Risk? Which is safer – staying home for the day or going to school/work? In 2003, one in 37 people was disabled for a day or more by an injury at home – more than in the workplace and car crashes combined Shave with razor – 33,532 injuries Hot water – 42,077 injuries Slice a grapefruit with a knife – 441,250 injuries

Which Has More Risk? What if you run down two flights of stairs to fetch the morning paper? 28% of the 30,000 accidental home deaths each year are caused by falls (poisoning and fires are the other top killers)

What Should We Do? Hide in a cave? Know the data – be aware!