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# Probability

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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
   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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