NCERT Class 11-Math's: Exemplar Chapter –16 Probability Part 2

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16.1.9 Mutually exclusive and exhaustive events:

If are n events of a sample space and if for every , i.e., and are pairwise disjoint and , then the events E1, E2, ... , En are called mutually exclusive and exhaustive events.

Consider the example of rolling a die.

We have

Let us define the three events as

A = a number which is a perfect square

B = a prime number

C = a number which is greater than or equal to 6

Now

Note that . Therefore, A, B and C are exhaustive events.

Also

Hence, the events are pairwise disjoint and thus mutually exclusive.

Classical approach is useful, when all the outcomes of the experiment are equally likely. We can use logic to assign probabilities. To understand the classical method consider the experiment of tossing a fair coin. Here, there are two equally likely outcomes - head (H) and tail (T). When the elementary outcomes are taken as equally likely, we have a uniform probability model. If there are k elementary outcomes in S, each is assigned the probability of . Therefore, logic suggests that the probability of observing a head, denoted by , is , and that the probability of observing a tail, denoted , is also . Notice that each probability is between 0 and 1.

Further H and T are all the outcomes of the experiment and .

16.1.10 Classical definition:

If all of the outcomes of a sample space are equally likely, then the probability that an event will occur is equal to the ratio:

Suppose that an event E can happen in ways out of a total of possible equally likely ways.

Then the classical probability of occurrence of the event is denoted by

The probability of non-occurrence of the event E is denoted by

Thus

The event ‘not is denoted by or (complement of E)

Therefore

16.1.11 Axiomatic approach to probability:

Let be the sample space of a random experiment. The probability P is a real valued function whose domain is the power set of S, i.e., and range is the interval i.e. satisfying the following axioms.

(i) For any event .

(ii)

(iii) If E and F are mutually exclusive events, then .

It follows from (iii) that .

Let be a sample space containing elementary outcomes ,

i.e.,

It follows from the axiomatic definition of probability that

(i)

(ii)

(iii) for any event A containing elementary events .

For example, if a fair coin is tossed once

satisfies the three axioms of probability

Now suppose the coin is not fair and has double the chances of falling heads up as compared to the tails, then .

This assignment of probabilities are also valid for H and T as these satisfy the axiomatic definitions.

16.1.12 Probabilities of equally likely outcomes:

Let a sample space of an experiment be and suppose that all the outcomes are equally likely to occur i.e., the chance of occurrence of each simple event must be the same i.e., for all , where

Since

Let be the sample space and E be an event, such that and . If each outcome is equally likely, then it follows that

16.1.13 Addition rule of probability:

If A and B are any two events in a sample space S, then the probability that atleast one of the events A or B will occur is given by

Similarly, for three events A, B and C, we have

16.1.14 Addition rule for mutually exclusive events:

If A and B are disjoint sets, then

, where A and B are disjoint].

The addition rule for mutually exclusive events can be extended to more than two events.

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