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Introduction to Probability and Risk 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. Subjective (personal) probability – use personal judgment or intuition. If you go to college today, you will be more successful in the future. 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? 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 Let’s try to solve these: ◦ 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 zip codes in the US? In Canada? 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) 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 What if we were to toss 2 coins? What are the theoretical probabilities of a two-coin toss? ◦ HH, HT, TH, TT – 4 possibilities, so each is 1/4 Now let’s toss 2 coins 10 times and observe the results (empirical results) Compare the theoretical results to the empirical 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 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) When you say “this occurring AND this occurring” you multiply the probabilities For example, suppose you toss three coins. What is the probability of getting three tails (getting a tail and a tail and a tail)? 1/2 x 1/2 x 1/2 = 1/8 (8 combinations of H and T, so each is 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 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? 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 When you say “this occurring OR that occurring”, you ADD the two probabilities What is the probability of something happening at least once? P(at least one event A in n trials) = 1 - [P(A not happening in one trial)]n 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 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 McDonalds has a new promotion. If you buy a large drink, your cup has a scratch off label on it. One in 20 cups wins a free Quarter Pounder. If you purchase 5 large drinks, what is the probability that you will win a Quarter Pounder? 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 is what you expect to gain or lose over the long run. 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 + … 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? 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 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. 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? 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 Let’s stop here for today. With terrorism, homicides, and traffic accidents, is it safer to stay home and take a college course online rather than head downtown to class? 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? 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? 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? 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? Instead compare deaths per trip? 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? Suppose you can purchase an extended warranty plan for a new auto which covers 100% of the engine and drive train (roughly 33% of the car) but no other items at all Or you can purchase an extended warranty plan which covers the entire auto but only at 33% coverage Which would you choose? 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 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 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) 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 Hide in a cave? Know the data – be aware! Now, let’s start our first med school lecture Welcome to the DePaul School of Medicine! Most people associate tumors with cancers, but not all tumors are cancerous Tumors caused by cancer are malignant Non-cancerous tumors are benign We can calculate the chances of getting a tumor and/or cancer – this is based on empirical research studies and probabilities If you don’t know how to calculate simple probabilities, you will misinform your patient and cause undo stress 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? Earlier mammogram screening was 85% accurate 85% would lead you to think that if you tested positive, there is a pretty good chance that you have cancer. But this is not true. Do the math! 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) Tumor is Tumor is Totals Malignant Benign Positive Mammogram Negative Mammogram Total 100 9900 10,000 Tumor is Malignant is 1/100th of the total 10,000. 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 Tumor is Tumor is Benign Totals Malignant Positive 85 True Mammogram Positives Negative 15 False Mammogram Negatives Total 100 9900 10,000 Mammogram screening correctly identifies 85% of the 9900 benign tumors as benign Thus it gives negative (benign) results for 85% of 9900, or 8415 These are called 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. Tumor is Tumor is Benign Totals Malignant Positive 85 True 1485 False Mammogram Positives Positives Negative 15 False 8415 True Mammogram Negatives Negatives Total 100 9900 10,000 This is what a mammogram should show: True Positives and True Negatives Tumor is Tumor is Benign Totals Malignant Positive 85 True 1485 False 1570 Mammogram Positives Positives Negative 15 False 8415 True 8430 Mammogram Negatives Negatives Total 100 9900 10,000 Now compute the row totals. 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, or 5.4% Thus, the chance that a positive result really means cancer is only 5.4% Therefore, when your patient’s mammogram comes back positive, you should reassure her that there’s still only a small chance that she has cancer 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? Tumor is Malignant Tumor is Benign Totals Positive 85 True 1485 False 1570 Mammogram Positives Positives Negative 15 False 8415 True 8430 Mammogram Negatives Negatives 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? That’s the end of the med school lecture for today.
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