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 years1 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!