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Probability
Introduction
Probability is the study of random situations.
It involves the chances of an event occurring.
Relative Frequency,
f of an event E in an
experiment is, the ratio of the number of occurrences 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
occurrences.
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.
A
Game with Random Numbers
Example
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}.