1. Math 4030-2a - Sample Space - Events - Definition of Probabilities
Math a - Sample Space - Events - Definition of Probabilities (of Events)
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2. Random Experiment An experiment is called Random experiment if
The outcome of the experiment in not known in advance
All possible outcomes of the experiment are known.
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3. Sample space and events
Set of all possible outcomes of an experiment is called sample space
We will denote a sample space by S
finite or infinite.
discrete or continuous.
Any subset of a sample space is called an event.
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4. Operations on events Mutually Exclusive Events Union, , “or”
Intersections,  , “and”
Complement, , “not”
Venn diagram
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5. How many outcomes in the sample space?
Tree diagram:
Identify the stages in the experiment;
Identify possible outcomes at every stage;
Count the number of “leaves” , for the size of the sample space.
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6. Multiplication Rule: k stages; there are n1 outcomes at the 1st stage;
from each outcome at ith stage, there are ni outcomes at (i+1)st stage; i=1,2,…,k-1.
Total number of outcomes at kth stage is
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7. Permutation Rule: n distinct objects;
take r (<= n) to form an ordered sequence;
Total number of different sequences is
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8. Factorial notation:
Permutation number when n = r, i.e.
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9. Combination Rule: n distinctive object;
take r (<= n) to form a GROUP (with no required order)
Total number of different groups is
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10. Count without counting:
Multiplication: Independency between stages;
Permutation: Choose r from n (distinct letters) to make an ordered list (words). Special case of multiplication;
Factorial: Special case of permutation;
Combination: Choose r from n, with no order.
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11. Probability of an event
P(A)
Event A S
[0,1]
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12. Axioms of probability:
Axiom 1. 0 ≤ P(A) ≤ 1. Axiom 2. P(S) = 1 Axiom 3. If A and B are mutually exclusive events then P(A U B) = P(A) + P(B)
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13. Axioms of probability
Generalization of Axiom 3. If A1, A2, …, An are mutually exclusive events in a sample space S then P(A1 U A2 U … U An) = P(A1) + P(A2) + … + P(An)
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14. Addition rule of probability
If A and B are any events in S then
P(AUB) = P(A) + P(B) – P(A  B)
Special case: if A and B are mutually exclusive, then
P(AUB) = P(A) + P(B).
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15. Classical probability has assumptions:
There are m outcomes in a sample space (as the result of a random experiment);
All outcomes are equally likely to occur;
An event A (of our interest) consists of s outcomes;
Then the definition of the probability for event A is
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16. Probability rule of the complement
If B is the complement of A, then
P(B) = 1 - P(A).
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17. Relative frequency approach
Perform the experiment (trial) m times repeatedly;
Record the number of experiments/trials that the desired event is observed, say s;
Then the probability of the event A can be approximated by
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