Page 204 - DMTH202_BASIC_MATHEMATICS_II

Unit 14: Probability

Notes
Example: A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single
marble is selected at random from the jar, what is the probability of selecting a red marble? a
green marble? a blue marble? a yellow marble?
The possible conclusions of this experiment are red, green, blue and yellow.
Probabilities:

# of ways to choose red 6 3
P(red) =  
total # of marbles 22 11
# of ways to choose green 5
P(green) = 
total # of marbles 22

# of ways to choose blue 8 4
P(blue) =  
total # of marbles 22 11
# of ways to choose yellow 3
P(yellow) = 
total # of marbles 22
The conclusions in this experiment are not equally expected to happen. You are more probable
to select a blue marble than any other color. You are least likely to select a yellow marble.
14.4.1 Axiomatic Approach to Probability

The issue of what probability actually is does not have a completely acceptable answer. In some
conditions it may be supportive to consider probability as displaying long-run amount or
degree of belief. However these are not exact mathematical definitions. The current approach is
to consider probability as a mathematical construction pleasing definite axioms.
Definition (Kolmogorov’s Axioms for Probability): Probability is a function P which allocates to
every event A a real number P(A) such that:
1. For every event A we have P(A)  0
2. P(S) = 1
3. If A1, A2, . . . , An are events and Ai  Aj =  for all i =6 j then
n
 
P  1A  A 2   An    P A i
i 1
If A1, A2, . . . are events and Ai  Aj = “ for all i =6 j then

 
P  1A  A 2     P A i
i 1
The events fulfilling 3 are pair wise disjoint or mutually exclusive.

Notes Notice that Axiom 3 has an edition for finitely numerous events and an edition for
a countable infinite number of events. If S is finite then we want only worry regarding the
first one of these. If S is infinite (mainly if it is not countable) then a number of intricacies
creep in which we will mostly ignore in this course.

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