Probability:
The probability of a given event is an expression of likelihood or chance of occurrence of an event. A probability is a number which ranges from zero to one, zero for an event which cannot occur and one for an event certain to occur.
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Definitions
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1. Experiment and outcomes:
Any operation on certain object or object which gives different results is called an ‘Experiment’ and the different possible results is called its ‘outcomes’.
Example:
Tossing of a coin is an experiment and appearance of head or tail are its outcomes.
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2. Event:
One outcome or a group of outcomes associated with certain conditions. It is called ‘Event’.
Example:
If one card is drawn from a pack of 52 playing cards, then drawing of: (i) Queen of heart; (ii) a card of spade; (iii) king of diamond etc., will be the examples of an event.
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3. Equally likely cases:
The cases are said to be equally likely when we have no reason to expect any one more rather than the other.
Example:
In a throw of an unbiased die any one out of the six faces may turn up so that all the six faces are equally likely.
4. Favourable Cases:
Out of all equally likely cases, those cases which fulfil the conditions of the ‘Event’ or the cases which entail the happening of the “Event” are known as favourable cases for that event.
Example:
If a die is rolled and the event is x > 4, then the favourable cases are two viz., x = 5 and x = 6, the remaining cases are unfavourable.
5. Exhaustive cases:
The total number of possible outcomes of a random experiment in a trial is known as the exhaustive number of cases.
Example:
In throwing of a die the exhaustive number of cases is 6, since any one of the six faces marked with 1, 2, 3, 4, 5, 6 may come uppermost.
6. Mutually exclusive events:
Events are said to be mutually exclusive or incompatible if the occurrence of any one of them prevents the occurrence of all the other i.e., if no two or more of them can occur simultaneously in the same trial.
7. Simple events:
The occurrence of a single event is known as the simple event.
8. Compound event:
Events obtained by combining together two or more elementary events are known as the compound events.
Example:
In a throw of a die the event getting a multiple of 2, is a compound event because this event occurs if any one of the elementary events 2, 4 or 6 occurs.
9. Independent event:
Events are said to be independent if the happening of one event is not affected by the happening of others.
Example:
If two cards are drawn from a well shuffled pack of 52 cards one after the other with replacement, then getting an ace in the first draw is independent to getting a king in the second draw but if the first card drawn in the first draw is not replaced, then second draw is dependent on the first draw.
Correlation:
If the change in one variable is accompanied by a change in the other variable in such a way that an increase in one variable results in an increase or decrease in the other and also greater change in one variable results in a corresponding greater change in other, then the two variables are said to be correlated.
Positive Correlation:
i. If the increase or decrease in one variable results in a corresponding increase or decrease in the other correlation is said to be positive.
ii. If the increase or decrease in one variable results in a corresponding decrease or increase, in other correlation is said to be negative.
iii. If the change in one variable is followed by corresponding and proportional change in other variable, correlation is said to be perfect.
The method to determine whether two variables are correlated or not
(i) Scatter Diagram Method —
In this method we use the rectangular coordinate axes to mark a dot corresponding to each pair of x and у values and thus obtain as many points as the number of ordered pairs in the given bivariate distribution. This diagram of dots is called the scatter diagram.
Regression:
If the variables in a bivariate distribution are correlated, then points in the scatter diagram generally condense around a curve, which we call the curve of regression and we say that there is a curve of linear regression between variables.
The Properties of Regression Coefficients —
1. The correlation coefficient and the two regression coefficients are of the same sign.
2. The correlation coefficient is the geometric mean between the regression coefficients.
3. Both the regression coefficients can not be numerically greater than unity.
4. Regression coefficients are independent of change of origin but not of scale.
Distributions:
(a) Binomial Distribution:
It is a very useful distribution for dealing with discontinuous variates. It is the distribution in which the different frequencies form a binomial series. Binomial series is obtained by expanding a binomial on the basis of the binomial theorem.
(q + p)n = qn + nC1qn-1p + nC2qn-2 p2 + … pn
(b) Normal Distribution:
Normal distribution is represented by following equation:
(c) Poisson Distribution:
Where, m = np
This expression is known as Poisson distribution.