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Introduction to Probability
Elementary properties of probability
• Definition: Probability is the likelihood or chance of an event occurring.
• Probability of an event is usually referred to as relative frequency
• Probability statements used frequently in biostatistics – e.g., we say that we are
90% probably sure that an observed treatment effect in a study is real; the
success probability of this surgery is only 10%; the probability of tumor to
develop is 50%......
THE RELATIVE FREQUENCY INTERPRETATION OF PROBABILITY
• We are interested in learning about the probability of some event to
occur in a certain process based on the number of actual occurrence
of that event relative to number of times that process repeated.
• For example, our process could be rolling a dice, and we are
interested in the probability in the event that the number on the dice
is equal to 6.
• I did this dice experiment 20 times. Each time I recorded the sum of
the two dice and got the following outcomes: results are below:
1 3 2 2 4 6 4 5 3 1 3 5 6 4 2 1 2 5
6 2
Experimental Probability Vs Theoretical
Probability
• Experimental Probability = 3/20
• Theorotical probability = 1/6
• The theoretical probability is what you expect to happen, but it isn't
always what actually happens.
• When n increases experimental probability approaches theoretical
probability. When n
∞, experimental probability = theoretical
probability.
In general, the probability of an event can be approximated by the
relative frequency , or proportion of times that the event occurs.
PROBABILITY (EVENT) is approximately = # of times event occurs/# of
experiments
THE SUBJECTIVE INTERPRETATION OF
PROBABILITY
• The relative frequency notion of probability is useful when the
process of interest, say tossing a coin, can be repeated many times
under similar conditions. But we wish to deal with uncertainty of
events from processes that will occur a single time. e.g. probability of
a success of a surgery, probability to get A in class.
• A subjective probability reflects a person's opinion about the
likelihood of an event. Different between people
Elementary properties of probability
1. Given some process (or experiment) with n mutually exclusive
outcomes (called events), E1, E2, E3, …, En, the probability of
any event Ei is assigned a nonnegative number. That is:
P( Ei )  0
• In other words, all events must have a probability of more than
or equal to zero (no negative probability)
2. The sum of probabilities of the mutually exclusive outcomes is
equal to 1 (exhaustiveness):
P( E1 )  P( E2 )  P( E3 )  ...............P( En )  1
Elementary properties of probability
3. Mutual Exclusiveness: Two or more events are said to be
mutually exclusive if the occurrence of any one of them
means the others will not occur (That is, we cannot have 2
events occurring at the same time)
Mutually exclusive events
H
Non-Mutually exclusive events
A
T
Example: Tossing a coin: the
outcome of head or tail is mutual
exclusive events. No way in a
certain toss you will have head
and tail at the same time. It is
either H or T
Example:
A
B
B