Basic Probability Theory
• Genetics has a large
random component
– Which allele does a parent transmit to their
offspring?
– Will an offspring display the disease if both
parents are carriers?
• Hence, an understanding of probability is
critical to a complete understanding of
genetics
• Genomics has become largely statistical in
nature
Overview
• Events and Probabilities
• Rules of probabilities
– Prob sum to one
– And or Or rules
• Conditional probability
– Conditional probabilities
– Bayes theorem
• Application: Disease relative risks
Introduction to Probability
•
Events
are possible outcomes of some random
processes. Examples of events:
– The genotype of a random individual is Bb
– You pass 320
– the weight of a random individual is less than 150
pounds
• We can define
the
probability of a particular
event
, say A, as the fraction of outcomes in
which event A occurs.
• Denote Probability of A by
Pr(A),
or
Prob(A)
Example:
Flipping a coin
• If you flip a coin once, the only possible
outcome is a heads or a tails
– Pr(Head) = 0.75 means that the chance is 75%
that the outcome of a random flip is a head
– Hence, Pr(Tail) = 1 Pr(head) = 0.25
– If you do 100,000 flips, you expect the number
of heads to be around 75,000 (give or take).
•
Probabilities can be counterintutive:
Birthday
problem
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View Full DocumentUse rules of Probability
• Rule 1: Pr(A) lies between 0 and 1 for all A
– Probabilities are between zero (never occur)
and one (always occur)
• Rule 2: The sum of probabilities of all
mutually exclusive (i.e., nonoverlapping)
events is one
.
– If there are n possible outcomes, Pr(1) + Pr(2) +
… + Pr(n) = 1
– Hence, Pr(1) = 1  [Pr(2) + … + Pr(n)]
– KEY:
Often easier to compute the probability
of the COMPLETEMENT of an event to
compute its probability
AND and OR Rules
•
AND rule
: If A and B are
independent
events
(knowledge of one event tells us
nothing about the other event), then the
probability that BOTH A and B occur is
– Pr(A and B) = Pr(A) Pr(B)
– Hence
AND = multiply probabilities
•
OR rule
:
If A and B are
exclusive events
(nonoverlapping), then the probability that
EITHER A or B occurs is
– Pr(A or B) = Pr(A) + Pr(B)
– Hence
OR = add probabilities
Example: Suppose we are rolling a fair
dice and flipping a fair coin
• What is the probability of rolling an even
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 Spring '07
 WEINERT
 Conditional Probability, Probability, Probability theory, Probability space

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