LSP 121

Introduction to Probabilityand 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

* Number of 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 possible outcomes are possible when you roll two dice?– 6 * 6

• How many possible outcomes for having three children?– 2 * 2 * 2

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 outcomes (ie. how many different three-course meals)?– 2*5*3

• A college offers 12 natural science classes, 15 social science classes, 10 English classes, and 8 fine arts classes. How many choices? – 12*15*10*8 = 1400

Possible Outcomes Continued

• A license plate has 7 digits, each digit being 0-9. How many possible outcomes?

• What if the license plate allows digits 0-9 and letters A-Z?

• How many area codes in the US?

* Three Types of Probability1. Theoretical, (aka “a priori”) probability – based on a model

in which all outcomes are equally likely. • Probability of a die landing on a 2 = 1/6• Probability of coin coming up tails = 1/2

2. 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.

3. Subjective (personal) probability – use personal judgment or intuition. • If you go to college today, you will be more successful in the future.• The Blackhawks have a 45% chance of winning the cup again next year.

Theoretical Probability• P(A) = (number of ways A can occur) / (total number of outcomes)

– Denominator = # of outcomes discussed in previous slides• Probability of a head landing in a coin toss?

– Numerator: head can occur 1 way – Denominator: 2 possible outcomes

• = 1/2• Probability of rolling a 7 using two dice?

– Num: 1/6, 2/5, 3/4, 4/3, 5/2, 6/1 (6 ways) – Denom: 36 outcomes

• = 1/6• Probability that a family of 3 will have two boys and one girl?

– Num: 3 possible ways of having 2 boys and 1 girl (BBG,BGB,GBB)– Denom: 8 possible outcomes: (BBB,BBG,BGB,BGG,GBB, GBG, GGB, GGG)

= 3/8

Empirical Probability

• Recall: Empirical probability is a probability based on observations or experiments

• Example: 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 times in previous 2000 years = 1 time every 500 years– So, probability = 1/500 = 0.002

* Probability of an Event Not Occurring

• P(not A) = 1 - P(A)• Seems simple, but turns out to be very useful,

so don’t forget this rule• 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– We will contrast this with combining probabilities

for events that are not independent

• For independent events, 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

• What is the probability of drawing an ace of diamonds and then an Ace of clubs from the deck?– 1/52 * 1/52? No: These events are not independent.

Combining Probabilities(for 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?

“OR” Probabilities(for 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(roll 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 20 years?

• Recall: P(at least one event ‘A’ in ‘n’ trials) = 1 - [ P(not A in one trial) ]n

• Probability of a flood in one “trial” (i.e. one year) is 1/100. So, the probability of no flood is 99/100.

• P(at least one flood in 20 years) = 1 - P(no flood in one year)20

= 1 - P(0.99 )20 = 0.87

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 event ‘A’ in ‘n’ trials) = 1 - [ P(not A in one trial) ]n

• P(at least one winner in 10 tickets) = 1 – ( 1- 0.1 )10 = 1-0.910

= 0.65

* Law of Large Numbers• Law of large numbers: Probability only applies over a large

number of trials. • The probability of tossing a coin and landing tails is 0.5. But

what if you toss it 5 times and you get HHHHH?– Could this happen? Of course!– But if you flipped 500 times, do you think you’ll get 500 heads?

Almost impossible• The “law of large numbers” tells you that if you toss the

coin many times, you should get approximately 50% tails.– The more times you flip the coin, the more likely you are to get

50% of each– The fewer number of times you flip the coin, the less likely you

are to get 50% of each

Expected Value

• Furthermore, what if you have multiple related events – each of which has its own probability? What is the expected value from the set of all possible events?

• We call this the ‘expected value’• Expected value = (event 1 value x prob of

event 1) + (event 2 value x prob of event 2) + (event 3 value * prob event 3) + etc …

Expected Value - Example

• Suppose that $1 lottery tickets have the following probabilities and values:– 1 in 5 win a free $1 ticket– 1 in 100 win $5– 1 in 100,000 to win $1000– 1 in 10 million to win $1 million

• What is the expected value of a lottery ticket?

Expected Value- 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

• Expected value = (-1*1) + (1*1/5) + (5*1/100) + (1000*1/100000) + (1000000*1/10000000)

• = - $0.64 • That is, every ticket costs you an average of 64 cents

– A positive value refers to money we gain– A negative value comes from money we spend. (e.g. The minus

1 in the very first term comes from the fact that a tickets costs us 1 dollar)

– Don’t forget that negative number!!!

Solution Continued

• Now sum all the products:

-$1 + 0.20 + 0.05 + 0.01 + 0.10 = -$0.64So, averaged over many tickets, you should expect to

lose $0.64 (on average) for each lottery ticket that you buy. If you buy, say, 1000 tickets, you will win with some of them and you will lose with some of them.

However over a 1000 tickets, you should expect to lose about $640.

Expected Value - Another Example

• 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.

• Does the company make or lose money on average? How much?

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 (Law of large numbers)– Recall that the law of large numbers tellls us that

probabilities are guaranteed to reach their expected values only when averaged over many trials.

Part II - RISK

• Uses probabilities…

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 (7 x 109) miles in the air• 100 deaths / 7 x 109 miles = 1.4 x 10-8 deaths per mile• How does this compare to driving (1.5 x 10-8 deaths per

mile)?• Is it fair to compare miles driven to miles flown?

– Might be more accurate to compare deaths per trip– Key point: Even when you come up with a nice statistical

number, it is no substitute for thinking. This is where many people (even those who should know better) drop the ball.

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 33% of the items completely but remaining items not at all

• Or you can purchase an extended warranty plan which covers all items at 33% 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)

Which Has More Risk?

• Ratio of people killed every year by lightning strikes versus number of people killed in shark attacks: 4000:1

• Average number of people killed worldwide each year by sharks: 6

• Average number of Americans who die every year from the flu: 36,000

What Should We Do?

• Hide in a cave?• Know the data – be aware!

• Now, let’s start our first med school lecture

Tumors and Cancer

• Welcome to the DePaul School of Medicine!• Most people associate tumors with cancers,

but not all tumors are cancerous• Tumors caused by cancer are referred to as

malignant• Non-cancerous tumors are referred to as

benign

Tumors and Cancer

• We can calculate the chances of getting a tumor and/or cancer. Our probability data is based on empirical research studies.

• If you don’t know how to calculate simple probabilities, you will misinform your patient and cause undo stress

Mammograms• Suppose your patient has a breast tumor.

Is it cancerous?– Probably not– Studies have shown that only about 1 in 100

breast tumors turn out to be malignant– Nonetheless, you order a mammogram– Suppose the mammogram comes back

positive. Now does the patient have cancer?

Accuracy

• Key Question: What is meant by a “positive” test?

• Earlier mammogram screening was 85% accurate– This might lead you to think that if you tested

positive, there is a pretty good chance that you have cancer.

• But this is not true!

Actual Results

• Consider a study in which mammograms are given to 10,000 women with breast tumors

• Assume that 1% (1 in 100) of the tumors are malignant (100 women actually have cancer, 9900 have benign tumors)

Actual Results

Tumor is Malignant

Tumor is Benign Totals

PositiveMammogram

NegativeMammogram

Total 100 9900 10,000

Tumor is Malignant is 1/100th of the total 10,000.

Actual Results

• Mammogram screening correctly identifies 85% of the 100 malignant tumors as malignant

• These are called true positives• The other 15% had negative results even

though they actually have cancer• These are called false negatives• There is a corresponding number of false

positives and true negatives

Possible Results from a Medical Test

• True Positive: patient has the disease, and test is positive

• False Positive: patient doesn’t have the disease, and test is positive

• True Negative: patient doesn’t have the disease, and test is negative

• False Negative: patient has the disease, and test is negative

• Pop-Quiz: Of these four, which is the most dangerous possibility?

Mammogram Results Tumor is Malignant Tumor is Benign Totals

PositiveMammogram

85 TruePositives

???

NegativeMammogram

15 FalseNegatives

???

Total 100 9900 10,000

Which labels (TP, FP, TN, FN) would be applied under the benign category?

Actual Results• Mammogram screening correctly identifies 85% of the

9900 benign tumors as benign “true negatives”

• The other 15% of the 9900 (1485) get positive results in which the mammogram incorrectly suggest their tumors are malignant. These are called false positives.

• Key point: One of the most common false assumptions made by non-medical folks, is to assume that tests are always accurate. – They are not– A good test has close to 100% accuracy, but very few

actually do

Actual Results

Tumor is Malignant Tumor is Benign Totals

PositiveMammogram

85 TruePositives

1485 FalsePositives

NegativeMammogram

15 FalseNegatives

8415 TrueNegatives

Total 100 9900 10,000

This is what a mammogram would ideally show: True Positives and True Negatives. Unfortunately, all tests have some error in them.

Actual Results

Tumor is Malignant Tumor is Benign Totals

PositiveMammogram

85 TruePositives

1485 FalsePositives

1570

NegativeMammogram

15 FalseNegatives

8415 TrueNegatives

8430

Total 100 9900 10,000

Now compute the row totals.

Results

• Overall, the mammogram screening gives positive results to 85 women who actually have cancer and to 1485 women who do not have cancer

• The total number of positive results is 1570• Because only 85 of these are true positives,

that is 85/1570, or 0.054• In other words, the chance that a positive

result really means cancer is only 5.4% !!

Teach your student doctor:

• When your patient’s mammogram comes back positive, you should reassure her that there’s still only a small chance that she has cancer – Although further tests are probably necessary

Another Question

• Suppose you are a doctor seeing a patient with a breast tumor. Her mammogram comes back negative. Based on the numbers above, what is the chance that she has cancer?– In other words, what is the probability that this is

a false negative? • Scary – but it happens! However, it is quite rare.

Actual Results

Tumor is Malignant

Tumor is Benign

Totals

PositiveMammogram

85 TruePositives

1485 FalsePositives

1570

NegativeMammogram

15 FalseNegatives

8415 TrueNegatives

8430

Total 100 9900 10,000

15/8430, or 0.0018, or slightly less than 2 in 1000.

This is a dangerous position. Now what do you do?Answer: Go to medical school to find out!