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PROBABILITY
Basic Concepts
So simple . . .
Figure these out
Take a blank piece of paper and write down
your own answers before they show up on
the slides.
At the Bus Stop
N 20
N 30
N 40
Chance that the next bus is N 30?
THE RULE
If events are equally likely
• Count the total number of possible events
• Count the total number of successful
events
• P(success) = # successful events
# total possible events
BUT … Suppose we know that
there are three times more
N40 than N20 or N30
N20, N30, N40, N40, N40
Probability that the next bus is
N30?
1/5
1 out of 5
20%
Take “weight” into consideration
Probability that the next bus will be N40
(N20), (N30) and (N40 X3)
3/5
60%
Note: this result is exactly the same as
counting all possibilities.
Definitions
These definitions make it possible to use
some shortcuts rather than always
counting individual events
DECK OF CARDS
This traditional deck of cards is also called a
Poker Deck. (There are others, e.g. a
Pinochle Deck.)
• 52 cards
• 13 each:
Clubs Diamonds Hearts Spades
• 4 each:
A 1 2 3 4 5 6 7 8 9 10 J Q K
Chance that the next card is 4◊
1/52
Chance that the next card is both
4◊ and A
IMPOSSIBLE!! Cannot be both!!
Mutually Exclusive Events
Mutually Exclusive Events
Both cannot occur together.
They cannot occur as a single event.
Not Mutually Exclusive
Chance of a card being a
Can pick a
Or a 7
Or a 7
and a 7.
Chance of a 7
= 1/52
OR
Chance of a 7 = 1/13
Chance of a heart = ¼
Chance of both together = 1/13 X ¼
= 1/52
2 or more events
not mutually exclusive
Probability that BOTH will occur equals
the product of the probability of each
occurring
Mary and Tom’s Baby
What is the chance that the baby will be a
boy?
What is the chance that the baby will have
blond hair?
What is the chance that the baby will be a
boy and have blond hair?
Chance that either one of two or
more events will occur
The shortcut depends on whether events
are mutually exclusive or not mutually
exclusive
To get to my house, I can take
either the N20 or the N30 bus
• There are three buses with an equal
chance of arriving on this route
N20, N30, N40.
• What is the chance that the next bus will
be either N20 or N30?
• Count them: answer is …
2/3
Are the two events mutually
exclusive?
Yes. The bus can’t be both N20 and N30.
The shortcut to get the answer is called
the “Addition Rule”.
The chance that the bus will be either
N20 or N30 is the sum of the
probability of N20 and the probability of
N30.
Sum of the Probabilities
P(N20) = 1/3
P(N30) = 1/3
P(N20 or N30) = Sum = 1/3 + 1/3 =
2/3
Shortcuts or rules always have the
same answer as the count-up.
Either a 7 or a
Are they mutually exclusive?
No
Can we still use the addition rule?
Yes, a more general form of the
addition rule
Either a 7 or a
Chance of a 7
= 1/13 = 4/52
Chance of a
= 1/4 = 13/52
Chance of either a 7 or a
Wrong!!!
= 4/52 + 13/52
Can be a 7 or a
or both
There is one card that we have counted
twice.
The 7
was counted in the 7’s and in
the
‘s.
We have to subtract one of these times
from the total.
P(7 or
) = 4/52 + 13/52 – 1/52
Chance of either this event or that
event
• If mutually exclusive, just add the
individual probabilities.
Addition Rule
• If not mutually exclusive, add the individual
probabilities and subtract the probability
that both will occur.
Generalized Addition Rule.
Why call it the Generalized Addition
Rule
• It applies to non-mutually exclusive events
• Does it apply to mutually exclusive
events?
• Try it for the buses: P(N20 or N30)
P(either N20 or N30) = P(N20) +
P(N30) – P(both)
= 1/3 + 1/3 – 0 = 2/3
Yes, the general rule applies to both Mutually exclusive just gives us
another shortcul.
Multiplication Rule
This applies to the chance that 2 or more
events will both occur.
What is the probability of the 10 ◊ ?
P (10) = 1/13
P (◊) = ¼
P (10 and ◊) = 1/13 X ¼ = 1/52
Same answer as counting-up.
Requirements for the
Multiplication Rule
Are the events independent or conditional?
The occurrence of a 10 and a ◊ are
independent of each other.
We can use the multiplication rule.
Mary & Tom’s Baby
• What is the chance that it will be a boy and
be blond?
Are they independent events?
P = p(boy) X p(blond)
• Two years later They have the first child
and are now expecting a second.
What is the chance that this child will be a
boy? Is it affected by the fact that the first
child was a boy?
Probability of two boys
• If we ask the question before they have
any children
P(2boys) = ½ X ½ = ¼
• If we ask the question after they already
have one boy
P(2nd boy) = 1/2
Conditional Events
• Class of 10
• 6 little boys: 2 blond, 1 black-haired, 3
brown-haired
• 4 little girls: 2 blond, 2 black hair
Principal asks me to send down one boy.
What is the chance that the child will have
blond hair?
2/6 = 1/3
Conditional Events cont’d
If the principal had asked me to send down
one child,
What is the probability that the child will be
blond?
4/10
Conditional Defined
• The chance of an event is dependent on
some other pre-existing condition, e.g. the
gender in this particular case determines
the probability of a hair color.
Chance that “both” events will
occur
The principal has asked that I send down a
child
• Chance that the child picked will be a boy
and blond.
• P(boy & blond) = P(boy) X P(blond|boy)
• P(blond|boy) reads “probability that the
child will be blond given that the child is a
boy”.
P(blond boy) from this class
• P(boy) = 6/10
• P(blond|boy) = 2/6
• P(blond boy) = 6/10 X 2/6 = 2/10 = 20%
• Look back and do count-up.
There are 2 blond boys out of the 10
children. P of a blond boy = 2/10 = 1/5
Another View
Boys
Girls
Total
Blond
2
2
4
Black hair
1
2
3
Brown hair
3
0
3
Total
6
4
10
Shortcuts & Definitions
Using a Table
All names in a hat
Probability of picking a girl
= 4/10
This is called a Marginal Probability.
Look back at table.
What’s the chance of picking a child with black
hair?
3/10
Is that a marginal probability?
Yes
Conditional Probability cont’d
• Use the table to find the probability of
choosing a brown-haired child from among
the girls
0/4 = 0
•Find the probability of choosing a
brown-haired child, given that the
child is a boy
3/6 = 2/3
Using the table just makes life
easier
Theoretical vs Experiential
• When we did chance of a boy in Mary and
Tom’s family, we used a theoretical
probability: 1 out of 2.
• When we did the class, we used an
experiential situation.
Mathematical Terms
• Union symbolizes either. It uses the
Addition Rule
U is the symbol for union
• Intersection symbolizes both. It uses the
Multiplication Rule
∩ is the symbol for intersection
Sum of the probabilities
• If we ask the probability that the next bus
will be either N20 or N30 or N40, the
probability will be 1/3 + 1/3 + 1/3 = 1.
• This is always the case. The total of the
probabilities is always 1.
Complementary
• Ask the probability that the next card is not
a 10 of diamonds.
• The probability that it is the 10 of
diamonds is 1/52 so the probability that it
is not the 10 of diamonds is 1 minus 1/52
= 51/52.
• That it is not the 10 of diamonds is the
complement of being the 10 of diamonds.
Replacement or Not Replacement
• We take a card and it is the 6 of spades.
• If we take a second card, the probability
that it will be the 4 of clubs depends on
whether we replace the first card or do not
replace it before the 2nd pick.
• With replacement, the chance of a 4 of
clubs is 1/52. Independent Events
• Without replacement, the chance of a 4 of
clubs is 1/51. Conditional Event.
Clinical Applications
• We can use our shortcuts to look at the
usefulness of a diagnostic test.
• Look at sensitivity and specificity
Diagnostic Test
Test Result Disease
Present
Disease
Absent
Total
Positive
436
5
441
Negative
14
495
509
Total
450
500
950
Sensitivity
• This is a conditional probability
• Given the presence of the disease, what is
the probability that the test will be positive
P = 436/450
Specificity
• This is a conditional probability
• Given that the disease is not present, what
is the probability that the test will be
negative
P = 495/500
Diagnostic Tests cont’d
• When someone has a positive result from
a screening test, what is it that they really
want to know.
• “Doc, I have a positive result, what does
this mean. Does it mean that I definitely
have the disease? If not, what is the
probability that I have the disease?
• In Statistics, this is called Predictive Value
Positive.
Predictive Value Positive
• Use Bayes’ Theorem to calculate it.
• Requires the conditional probabilities used
for specificity and sensitivity
• Also requires the prevalence of the
disease in the general population
Calculate
D1 = probability that an individual has the disease
D2 = the complement
T* = test is positive
P(D1|T*) = P(D1∩T*)
P(T*)
Equals
P(D1)P(T*|D1)
P(D1)P(T*|D1) + P(D2)P(T*|D2)
OOOPS!
The table doesn’t give us P(D1); have to get that from the population
Something we don’t know!!!
P(D1) is the rate of the disease in the
general population,
Called the prevalence of the disease,
We don’t get this from the table because it
only contains info on those tested not the
whole population.
Example:
probability that a woman has cervical cancer
In a certain year was 83 per million
= 0.000083
The End
of Power Pt 1, Probability
Now, try some of the problems together.
Go to end of Ch. 6