Probability Introduction
Probability is the study of random situations. It involves the chances of an event occuring.

Relative Frequency
Relative frequency f of an event E in an experiment is, the ratio of the number of occurences of that particular event to the number of times the experiment is performed.

For a die tossed N times, let s be the number of times an ace occurs,                                           
                                           f(E) = s/N.

Probability of the event P(E) is an ideal ratio depicting the relative frequency. The probability experiment is a specific set of actions in which the results are not able to be predicted with certainty.

Sample Space
If each outcome of an experiment is represented by a unique element and each element represents a unique outcome, the set of outcomes S for that particular experiment is called a sample space. The sample space is determined by what an observer chooses to record. More than one sample space is possible for a given experiment. The best way to determine a sample space for any particular experiment is to first figure out the likely occurences.

Eg.   
        A coin is tossed twice.
             (a) Indicate a sample space for this experiment.
             (b) Indicate also a sample space for the number of heads obtained.

Solution
(a) A coin has two sides, heads and tails. If the coin is tossed once, then  either H (heads) or T (tails) could occur. If it is tossed twice, both heads could occur. Both tails could occur also. Likewise, either heads or tails could occur on either toss. The appropriate sample space S for this experiment would be the set of possible outcomes, being S = {TT, HH, TH, HT}. Both letters together represent the possibilities of the first and second toss.
(b) For the number of heads, there could be 1 (one) on the first toss and 1 on the second, making it 2 heads. There could also be 1 on the first and 0 (none) on the second. Vice versa, there could be 0 on the first and 1 on the second. There could even be 0 on both tosses. The sample space then for this would be S = {2, 1, 0}.

Events
Previously we mentioned that a sample space is a set of possible outcomes. An event in probability is a subset to a special kind of set called a sample space. From the experiment of tossing a coin twice, recall that the sample space was S = {TT, TH, HT, HH}. The event of having heads on the second toss would be {TH, HH}. The event of having heads on both tosses would be {HH}. The event of having at least one tail would be {TT, TH, HT} and so on.

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