probability

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Probability
Introduction to Probability,
Conditional Probability

2. Introduction
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• Probability is the study of randomness and
uncertainty of any outcome.
• In the early days, probability was associated with
games of chance (gambling).

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Simple Games Involving Probability
Game: A fair die is rolled. If the result is 2, 3, or 4, you win
$1; if it is 5, you win $2; but if it is 1 or 6, you lose $3.
Should you play this game?

4. Why do we need Probability?
• We have several graphical and numerical
statistics for summarizing our data
• We want to make probability statements
about the significance of our statistics
• Eg. In MSPH class, mean (height) = 5.4 feet
• What is the chance that the true height of MSPH
students is between 5 feet and 6 feet ?

5. Deterministic vs. Random Processes
• In deterministic processes, the outcome can be
predicted exactly in advance
• Eg. Force = mass x acceleration. If we are given
values for mass and acceleration, we exactly know
the value of force
• In random processes, the outcome is not
known exactly, but we can still describe the
probability distribution of possible outcomes
• Eg. 10 coin tosses: we don’t know exactly how
many heads we will get, but we can calculate the
probability of getting a certain number of heads

6. Random Experiment…
• …a random experiment is an action or
process that leads to one of several possible
outcomes/events. For example:
Experiment Outcomes
Flip a coin Heads, Tails
Exam Marks Numbers: 0, 1, 2, ..., 100
Assembly Time t > 0 seconds
Course Grades F, D, C, B, A, A+

7. Events
• An event is an outcome or a set of outcomes of
a random process/experiment
Example: Tossing a coin three times
Event A = getting exactly two heads = {HTH, HHT, THH}
Example: Picking real number X between 1 and 20
Event A = chosen number is at most 8.23 = {X ≤ 8.23}
Example: Tossing a fair dice
Event A = result is an even number = {2, 4, 6}
• Notation: P(A) = Probability of event A

8. Sample Space
• The sample space S of a random process is
the set of all possible outcomes
Example: one coin toss
S = {H,T}
Example: three coin tosses
S = {HHH, HTH, HHT, TTT, HTT, THT, TTH, THH}
Example: roll a six-sided dice
S = {1, 2, 3, 4, 5, 6}
Example: Pick a real number X between 1 and 20
S = all real numbers between 1 and 20

9. Rules (Axioms) of Probability
Rule 1: 0 ≤ P(A) ≤ 1 for any event A
Rule 2: The probability of the whole sample space is 1
P(S) = 1
Rule 3: P(Ac
) = 1 - P(A)
Rule 4: If A and B are disjoint events then
P(A or B) = P(A) + P(B)
Rule 5: If A and B are independent
P(A and B) = P(A) x P(B)

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Combinations of Events
• The complement Ac
of an event A is the event that A
does not occur
• Probability Rule 3:
P(Ac
) = 1 - P(A)
• The union of two events A and B is the event that
either A or B or both occurs
• The intersection of two events A and B is the event
that both A and B occur
Event A Complement of A Union of A and B Intersection of A and B

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Disjoint Events
• Two events are called disjoint if they can not
happen at the same time
• Events A and B are disjoint means that the
intersection of A and B is zero
• Example: coin is tossed twice
• S = {HH,TH,HT,TT}
• Events A={HH} and B={TT} are disjoint
• Events A={HH,HT} and B = {HH} are not disjoint

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Independent events
• Events A and B are independent if knowing that A
occurs does not affect the probability that B occurs
• Example: tossing two coins
Event A = first coin is a head
Event B = second coin is a head
• Disjoint events cannot be independent!
• If A and B can not occur together (disjoint), then knowing that
A occurs does change probability that B occurs
• Probability Rule 5: If A and B are independent
P(A and B) = P(A) x P(B)
Independent
multiplication rule for independent events

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Equally Likely Outcomes Rule
• If all possible outcomes from a random process
have the same probability, then
• P(A) = (# of outcomes in A)/(# of outcomes in S)
• Example: One Dice Tossed
P(even number) = |2,4,6| / |1,2,3,4,5,6|

15. The AND Rule of Probability
• The probability of 2 independent events both happening
is the product of their individual probabilities.
• Called the AND rule because “this event happens AND
that event happens”.
• For example, what is the probability of rolling a 2 on one
die and a 2 on a second die? For each event, the
probability is 1/6, so the probability of both happening is
1/6 x 1/6 = 1/36.
• Note that the events have to be independent: they can’t
affect each other’s probability of occurring. An example
of non-independence: you have a hat with a red ball and
a green ball in it. The probability of drawing out the red
ball is 1/2, same as the chance of drawing a green ball.
However, once you draw the red ball out, the chance of
getting another red ball is 0 and the chance of a green
ball is 1.

16. The OR Rule of Probability
• The probability that either one of 2
different events will occur is the sum of
their separate probabilities.
• For example, the chance of rolling
either a 2 or a 3 on a die is 1/6 + 1/6 =
1/3.

17. NOT Rule
• The chance of an event not happening
is 1 minus the chance of it happening.
• For example, the chance of not getting
a 2 on a die is 1 - 1/6 = 5/6.
• This rule can be very useful.
Sometimes complicated problems are
greatly simplified by examining them
backwards.

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Conditional Probabilities
• The notion of conditional probability can be
found in many different types of problems
• Eg. imperfect diagnostic test for a disease
• What is probability that a person has the
disease? Answer: 40/100 = 0.4
• What is the probability that a person has the
disease given that they tested positive?
More Complicated !
Disease + Disease - Total
Test + 30 10 40
Test - 10 50 60
Total 40 60 100

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Definition: Conditional Probability
• Let A and B be two events in sample space
• The conditional probability that event B occurs
given that event A has occurred is:
P(A|B) = P(A and B) / P(B)
• Eg. probability of disease given test positive
P(disease +| test +) = P(disease + and test +) / P(test +) = (30/100)/(40/100) =.75

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