If a and B Are Two Events Associated with an Experiment Then

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PROBABILITY

(I) CONDITIONAL PROBABILITY

If A and B are two events associated with an experiment then

(1) Probability of occurrence of event A when event B has already occurred is denoted by

(2) Probability of occurrence of event A when event B has already occurred is denoted by

RESULTS

1. , where ‘S’ is the sample space and hence a sure event

2. If A and B are events of a sample space ‘S’ and ‘F’ is an event of the sample space such that then

3.

(II) MULTIPLICATION THEOREM

(1) Multiplication theorem for the conditional events: If A and B are two events then probability of simultaneous occurrence of events A and B is and given by

GENERALISATION

If are ‘n’ events such that they occur one after the other in the order then the probability of their simultaneous occurrence is and it is given by

In particular if A,B and C are three events appearing one after the other in the order then the probability of their occurrence is given by

(2) Multiplication theorem for independent events:

Independent events: Two or more events are said to be independent iff the occurrence of one have no effect on the occurrence of the other.

If A and B are two independent events then the probability of their simultaneous occurrence is given by

GENERALISATION

The simultaneous occurrence of independent eventsis given by

NOTE:

1. Two events A and B will be independent iff.

2. Eventswill be independent iff .

(III) LAW OF TOTAL PROBABILTY

If are ‘n’ mutually exclusive and exhaustive events and if A is the event that can occur with either of the events ,where then the probability of the occurrence of event A is given by

(IV) BAYE’S THEOREM

If are ‘n’ mutually exclusive and exhaustive events and if A is the event that can occur with either of the events ,where then the probability of the occurrence of event when event A has already occurred is given by

Where

(V) RANDOM VARIABLE AND PROBABILTY DISTRIBUTION

RANDOM VARIABLE

A random variable is a subset of real number that can be associated with the sample space of an experiment.

A random variable is denoted by capital English alphabets like X, Y, Z etc.

EXAMPLES:

1. In a throw of a coin we may take the random variable as

X = Number of tails appearing in the throw = 0,1

2. In a throw of a pair of coins we may take the random variable as

X = Number of heads appearing in the throw = 0,1,2

3. In a throw of a dice we may take the random variable as

X = Number appearing on the dice = 1,2,3,4,5,6

Y = Number of times even number appearing on it = 0,1

4. In a throw of a pair of dice we may take the random variable as

X = Number of times odd number appearing on it = 0,1,2

5. In a throw of a pair of dice we may take the random variable as

X = Sum of numbers appearing on the dice = 2,3,4,5,6,7,8,9,10,11,12

(VI) PROBABILTY DISTRIBUTION OF THE RANDOM VARIABLE AND ITS MEAN AND VARIANCE

PROBABILTY DISTRIBUTION

Probability distribution of the random variable is the all possible probabilities of the values of a random variable

i.e. If a random variable X has all possible values as then the probabilities

of the valuesrespectively of the random variable X are said to form the probability distribution of the random variable X.

NOTE:

1. Here

2. If are the probabilities distribution of the values of a random variable ‘X’ then

(VII) MEAN VARIANCE AND STANDARD DEVIATION OF A PROBABILTY DISTRIBUTION

If of the valuesrespectively of the random variable X then the

(1) Mean of the random variable is

(2) Variance of the random variable is

(3) Standard deviation of the random variable is

(VIII) BERNOULLI’S TRIAL

Bernoulli’s Trial: Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions:

(i) There should be a finite number of trials.

(ii) The trials should be independent.

(iii) Each trial has exactly two outcomes: success or failure.

(iv) The probability of success remains the same in each trial.

For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each

trial results in success (say an even number) or failure (an odd number) and the

Probability of success (p) is same for all 50 throws. Obviously, the successive throws

of the die are independent experiments. If the die is fair and have six numbers 1 to 6

written on six faces, then and

(V) BINOMIAL DISTRIBUTION

If the trials of a random variable are Bernoulli’s trial with probability of success as ‘p’ and probability of failure as ‘q’ then

Where

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