Probability Fundamentals
Definitions
Sample Space and Event: Consider an experiment whose outcome is not predictable with certainty. Such an experiment is called a random experiment. However, although the outcome of the experiment will not be known in advance, let us suppose that the set of all possible outcomes is known. This set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by S. Some examples follow.
1. If the outcome of an experiment consists in the determination of the sex of a newborn child, then S = {g, b} where the outcome g means that the child is a girl and b is the boy.
2. If the outcome of an experiment consists of what comes up on a single dice, then S = {1, 2, 3, 4, 5, 6}.
3. If the outcome of an experiment is the order of finish in a race among the 7 horses having post positions 1, 2, 3, 4, 5, 6, 7; then S = {all 7! permutations of the (1, 2, 3, 4, 5, 6, 7)}.
The outcome (2, 3, 1, 6, 5, 4, 7) means, for instance, that the number 2 horse comes in first, then the number 3 horse, then the number 1 horse, and so on.
Any subset E of the sample space is known as Event. That is, an event is a set consisting of some or all of the possible outcomes of the experiment. For example, in the throw of a single dice S = {1, 2, 3, 4, 5, 6} and some possible events are
E1 = {1, 2, 3}
E2 = {3, 4}
E3 = {1, 4, 6}, etc.
If the outcome of the experiment is contained in E, then we say that E has occurred. Always E ⊆ S. Since E & S are sets, theorems of set theory may be effectively used to represent and solve probability problems which are more complicated.
Example: If by throwing a dice, the outcome is 3, then events E1 and E2 are said to hare occurred.
In the child example – (i) If E1 = {g}, then E1 is the event that the child is a girl.
Similarly, if E2 = {b}, then E2 is the event that the child is a boy. These are examples of Simple events.
Compound events may consist of more than one outcome. Such as E = {1, 3, 5} for an experiment of throwing a dice. We say event E has happened if the dice comes up 1 or 3 or 5.
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