A coin comes down heads 50% of the time, on average. After many tosses,
the number of heads (#heads) is approximately equal to the number of tails.
In the limit as # tosses -> infinity
We say the probability of getting a head at any one toss is 1/2. This illustrates the concept of probability that will be used in this course.
John Kerrich, a South African mathematician, was visiting Copenhagen when World War II broke out. Two days before he was scheduled to fly to England, the Germans invaded Denmark. Kerrich spent the rest of the war interned at a camp in Jutland and to pass the time he carried out a series of experiments in probability theory. In one, he tossed a coin 10,000 times. His results are shown in the following graph.
(The horizontal axis is on a log scale)
The tossing of a coin 10 times is an example of a random experiment. Most experiments are subject to Random Variation. Probability theory is the mathematical approach to quantifying uncertainty or variation.
Toss a coin twice and record for each toss whether the result was a head (H) or a tail (T). Exercise: List the possible outcomes.
Let A be the event of one or more heads. Which outcomes belong to event A? (HT, TH, HH)
Let B be the event that there are no heads. (TT)
In this example, events A and B are said to be disjoint or mutually exclusive, as they have no outcomes in common. They are also exhaustive, as they cover all possible outcomes.
Exercise: Define an event C which is not disjoint from A.
An event is a set of one or more outcomes in the sample space.
Two events are disjoint or mutually exclusive if they have no outcomes in common.
Random variation occurs when it is impossible to predict with certainty the exact outcome of an individual experiment, but as the experiment is repeated a large number of times a regular distribution of relative frequencies emerges.
Probability of an outcome or event can be determined either empirically (based on data) or theoretically based on a mathematical model of the process. Empirical definition is as follows: Suppose an outcome (or event) A occurs f times in n observations. Then the
The concept of the probability of an event A is an idealisation of
relative frequency. It is the limiting value of relative frequency as n
becomes very large, i.e. as n -> 
.
(P(A) denotes the probability of A occuring)
Theoretical estimates of probability are based on plausible assumptions. The most common assumption is that all possible outcomes are equally likely. Then
By analogy with relative frequencies, probabilities have the following properties:
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