LSP 121

1
LSP 121

• Introduction to Probability
• and Risk

2
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.

3
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.

4
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?

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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

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Possible Outcomes Continued

• Lets 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?

7
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)

8
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

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Theoretical vs. Empirical

• 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 lets toss 2 coins 10 times and observe the
  results (empirical results)
• Compare the theoretical results to the empirical

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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

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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)
• When you say this occurring AND this occurring
  you multiply the probabilities

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Combining Probabilities -Independent Events

• 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

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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 Gamblers Fallacy
• Is this the same for playing Blackjack?

14
Either/Or Probabilities -Non-Overlapping Events

• If you ask what is the probability of either this
  happening OR that happening, and the two events
  dont 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

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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(A
  not happening in one trial)n

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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

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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

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You Try One

• 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?

19
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 is what you expect to gain or lose
  over the long run.

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Expected Value

• 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
  

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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?

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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

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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.

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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?

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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
• Lets stop here for today.

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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?

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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?

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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?

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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?

30
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? Instead compare deaths per trip?

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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?

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The Certainty Effect

• 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?

33
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

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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

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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)

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Which Has More Risk?

• Ratio of people killed every year by lightning
  strikes versus number of people killed in shark
  attacks 40001
• Average number of people killed worldwide each
  year by sharks 6
• Average number of Americans who die every year
  from the flu 36,000

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What Should We Do?

• Hide in a cave?
• Know the data be aware!
• Now, lets start our first med school lecture

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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 malignant
• Non-cancerous tumors are benign

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Tumors and Cancer

• We can calculate the chances of getting a tumor
  and/or cancer this is based on empirical
  research studies and probabilities
• If you dont know how to calculate simple
  probabilities, you will misinform your patient
  and cause undo stress

40
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?

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Accuracy

• 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!

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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)

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Actual Results
Tumor is Malignant Tumor is Benign Totals
Positive Mammogram
Negative Mammogram
Total 100 9900 10,000
Tumor is Malignant is 1/100th of the total 10,000.
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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

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Actual Results
Tumor is Malignant Tumor is Benign Totals
Positive Mammogram 85 True Positives
Negative Mammogram 15 False Negatives
Total 100 9900 10,000
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Actual Results

• 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.

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Actual Results
Tumor is Malignant Tumor is Benign Totals
Positive Mammogram 85 True Positives 1485 False Positives
Negative Mammogram 15 False Negatives 8415 True Negatives
Total 100 9900 10,000
This is what a mammogram should show True
Positives and True Negatives
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Actual Results
Tumor is Malignant Tumor is Benign Totals
Positive Mammogram 85 True Positives 1485 False Positives 1570
Negative Mammogram 15 False Negatives 8415 True Negatives 8430
Total 100 9900 10,000
Now compute the row totals.
49
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, or 5.4

50
Results

• Thus, the chance that a positive result really
  means cancer is only 5.4
• Therefore, when your patients mammogram comes
  back positive, you should reassure her that
  theres still only a small chance that she has
  cancer

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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?

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Actual Results
Tumor is Malignant Tumor is Benign Totals
Positive Mammogram 85 True Positives 1485 False Positives 1570
Negative Mammogram 15 False Negatives 8415 True Negatives 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? Thats the end of the med school lecture
for today.