Chapter 14 Probability
Where a mathematical reasoning can be had, it is as great a folly to make use of any other, as to grope for a thing in the dark, when you have a candle in your hand. - JOHN ARBUTHNOT
14.1 Event
We have studied about random experiment and sample space associated with an experiment. The sample space serves as an universal set for all questions concerned with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space is
Now suppose that we are interested in those outcomes which correspond to the occurrence of exactly one head. We find that
We know that the set
The above discussion suggests that a subset of sample space is associated with an event and an event is associated with a subset of sample space. In the light of this we define an event as follows.
Definition Any subset
14.1.1 Occurrence of an event
Consider the experiment of throwing a die. Let
Thus, the event
14.1.2 Types of events
Events can be classified into various types on the basis of the elements they have.
1. Impossible and Sure Events The empty set
To understand these let us consider the experiment of rolling a die. The associated sample space is
Let
Clearly no outcome satisfies the condition given in the event, i.e., no element of the sample space ensures the occurrence of the event
Now let us take up another event
2. Simple Event If an event
In a sample space containing
For example in the experiment of tossing two coins, a sample space is
There are four simple events corresponding to this sample space. These are
3. Compound Event If an event has more than one sample point, it is called a Compound event
For example, in the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc.
are all compound events. The subsets of
Each of the above subsets contain more than one sample point, hence they are all compound events.
14.1.3 Algebra of events
In the Chapter on Sets, we have studied about different ways of combining two or more sets, viz, union, intersection, difference, complement of a set etc. Like-wise we can combine two or more events by using the analogous set notations.
Let A, B, C be events associated with an experiment whose sample space is S.
1. Complementary Event For every event A, there corresponds another event
For example, take the experiment ‘of tossing three coins’. An associated sample space is
Let
Clearly for the outcome HTT, the event A has not occurred. But we may say that the event ’not A’ has occurred. Thus, with every outcome which is not in A, we say that ’not A’ occurs.
Thus the complementary event ’not A’ to the event A is
or
2. The Event ‘A or B’ Recall that union of two sets A and B denoted by A
When the sets
3. The Event ‘A and B’ We know that intersection of two sets
If
Thus,
For example, in the experiment of ’throwing a die twice’ Let
so
Note that the set
4. The Event ‘A but not B’ We know that A-B is the set of all those elements which are in A but not in B. Therefore, the set A-B may denote the event ‘A but not B’. We know that
14.1.4 Mutually exclusive events
In the experiment of rolling a die, a sample space is
Clearly the event A excludes the event B and vice versa. In other words, there is no outcome which ensures the occurrence of events A and B simultaneously. Here
Clearly
In general, two events
Again in the experiment of rolling a die, consider the events A ‘an odd number appears’ and event
Obviously
Now
Therefore, A and B are not mutually exclusive events.
Remark Simple events of a sample space are always mutually exclusive.
14.1.5 Exhaustive events
Consider the experiment of throwing a die. We have
A: ‘a number less than 4 appears’,
B: ‘a number greater than 2 but less than 5 appears’
and C: ‘a number greater than 4 appears’.
Then
Such events
then
Further, if
We now consider some examples.
14.2 Axiomatic Approach to Probability
In earlier sections, we have considered random experiments, sample space and events associated with these experiments. In our day to day life we use many words about the chances of occurrence of events. Probability theory attempts to quantify these chances of occurrence or non occurrence of events.
In earlier classes, we have studied some methods of assigning probability to an event associated with an experiment having known the number of total outcomes.
Axiomatic approach is another way of describing probability of an event. In this approach some axioms or rules are depicted to assign probabilities.
Let
(iii) If
It follows from (iii) that
Let
It follows from the axiomatic definition of probability that
(i)
(ii)
(iii) For any event
Note - It may be noted that the singleton
For example, in ‘a coin tossing’ experiment we can assign the number
i.e.
Clearly this assignment satisfies both the conditions i.e., each number is neither less than zero nor greater than 1 and
Therefore, in this case we can say that probability of
If we take
Does this assignment satisfy the conditions of axiomatic approach?
Yes, in this case, probability of
We find that both the assignments (1) and (2) are valid for probability of
In fact, we can assign the numbers
This assignment, too, satisfies both conditions of the axiomatic approach of probability. Hence, we can say that there are many ways (rather infinite) to assign probabilities to outcomes of an experiment. We now consider some examples.
14.2.1 Probability of an event
Let
A sample space associated with this experiment is
where
Let the probabilities assigned to the outcomes be as follows
Let event A: there is exactly one defective pen and event B: there are atleast two defective pens.
Hence
Now
and
Let us consider another experiment of “tossing a coin “twice”
The sample space of this experiment is
Let the following probabilities be assigned to the outcomes
Clearly this assignment satisfies the conditions of axiomatic approach. Now, let us find the probability of the event
Here
Now
For the event
and
14.2.2 Probabilities of equally likely outcomes
Let a sample space of an experiment be
Let all the outcomes are equally likely to occur, i.e., the chance of occurrence of each simple event must be same.
i.e.
Let
14.2.3 Probability of the event ’ or '
Let us now find the probability of event ‘A or B’, i.e.,
Let
Clearly
Now
If all the outcomes are equally likely, then
Also
and
Therefore
It is clear that
The points HTH and THH are common to both A and B. In the computation of
i.e.
Thus we observe that,
In general, if
Since
we have
(because
Also
Hence
Alternatively, it can also be proved as follows:
and
Subtracting (3) from (2) gives
The above result can further be verified by observing the Venn Diagram (Fig 14.1)
If
Thus, for mutually exclusive events
which is Axiom (iii) of probability.
14.2.4 Probability of event ’not ’
Consider the event
If all the outcomes
Now
Also event ’ not
Now
Thus,
Also, we know that
or
Now
or
We now consider some examples and exercises having equally likely outcomes unless stated otherwise.
Summary
In this Chapter, we studied about the axiomatic approach of probability. The main features of this Chapter are as follows:
Event: A subset of the sample space
Impossible event : The empty set
Sure event: The whole sample space
Complementary event or ’not event’ : The set
Event A or B: The set A
Event
Event
Mutually exclusive event:
Exhaustive and mutually exclusive events: Events
(i)
(ii)
(iii)
Equally likely outcomes: All outcomes with equal probability
Probability of an event: For a finite sample space with equally likely outcomes Probability of an event
If
If
If
Historical Note
Probability theory like many other branches of mathematics, evolved out of practical consideration. It had its origin in the 16th century when an Italian physician and mathematician Jerome Cardan (1501-1576) wrote the first book on the subject “Book on Games of Chance” (Biber de Ludo Aleae). It was published in 1663 after his death.
In 1654, a gambler Chevalier de Metre approached the well known French Philosopher and Mathematician Blaise Pascal (1623-1662) for certain dice problem. Pascal became interested in these problems and discussed with famous French Mathematician Pierre de Fermat (1601-1665). Both Pascal and Fermat solved the problem independently. Besides, Pascal and Fermat, outstanding contributions to probability theory were also made by Christian Huygenes (16291665), a Dutchman, J. Bernoulli (1654-1705), De Moivre (1667-1754), a Frenchman Pierre Laplace (1749-1827), the Russian P.L Chebyshev (18211897), A. A Markov (1856-1922) and A. N Kolmogorove (1903-1987). Kolmogorov is credited with the axiomatic theory of probability. His book ‘Foundations of Probability’ published in 1933, introduces probability as a set function and is considered a classic.