Subject: Mathematics
Permutation and Combination:
Permutation: Permutation of a set of objects means the arrangement of objects in some order.
Combination: Combination of a set of objects means the selection of objects without regard to any of the arrangements.
$$\;1.\;^nC_r\;=C(n,r)\;=\;\frac{n!}{(n-r)!\;r!\;}\;(r<n)$$ $$\;2.\;^nC_r\;=\;^nC{n-r}$$
Definitions of Probability:
Probability in ordinary languagesignifies chance. We talk about the probability of raining, the probability of an accident, the probability of success etc. To measure probability in a mathematical way, we need quantitative measures related to its outcomes. Probability can be measured using its classical, empirical and axiomatic definitions :
Classical or Priori Definition of Probability:
In an experiment, if there are ‘n’ exhaustive, mutually exclusive and equally likely case and m of them are favourable to an event ‘É’, then the probability of the happening of an event E is denoted by P(E) is defined by $$\;P(E)\;=\;p\;=\;\frac{m}{n}$$Here, p is often used for the value of probability.
The probability P(E) of happening of an event E satisfies the following properties :
Odds in favour and Odds against :
Sometimes probability is expressed in terms of odds in favour, and odds against an event. The odds in favour of an event is defined as proportion m:(n-m) and odd against E is (n-m):m. where n is the total number of cases and m is the total number of favourable cases.
Limitations of the Classical Definition of Probability
The classical definition of probability has the following limitations :
Empirical Definition of Probability
The empirical definition (also called the statistical definition or definition of relative frequency ) is as following.
If trials are to be repeated a greater number of time ( infinitely ), under the same conditions, then the ratio of the number of times the favourable events occurs to the total number of trials, as the number increases infinitely, is the probability of the event. If an event E occurs m times in n repetitions of a random experiment then the value approached by the relative frequency m/n when n is infinitely large is said to be the probability of an event E. Symbolically,
$$\;P(E)\;=\;\lim_{n\to\infty}\frac{m}{n}$$ provided that limit exists.
Axiomatic probability:
A.N Kolmogorov has introduced the concept of probability with thehelp of set theory. According to this definition:
If S is the sample space of a random experiment with number of sample points n(S) and the number of sample points favourable to an event E be n(E), then the probability of happening of an event P(E) is given by:
$$\;P(E)\;=\;\frac{n(E)}{n(S)}$$
The probability of happening of an event P(E) satisfies the following conditions:
Dependent events : Two events are said to be dependent if the occurrence of one event in a trial affects the probability of the occurrence of the other event in another trial.
For example : A bag contains 6 white balls and 4 red balls. A ball is drawn and it is a white ball. We find its probability.
n=total number of possible cases = 6 + 4 = 10
m = no. of favourable cases = 6
$$\;P(a\;white\;ball)\;=\;\frac{m}{n}\;=\;\frac{6}{10}\;=\;\frac{3}{5}$$
If the ball drawn is not replaced, then the balls left are 5 white balls and 4 red balls. Again, we find the probability of getting a white ball.
n=total number of possible cases = 5 + 4 = 9
m = no. of favourable cases = 5
$$\;P(a\;white\;ball)\;=\;\frac{m}{n}\;=\;\frac{5}{9}\;which\;is\;different\;from\;\frac{3}{5}$$
Hence, the probability of getting a second white depends on the occurrence of the 1^{st} white ball. Hence the event of drawing the balls one by one where the first ball is not replaced before the second is drawn, are dependent.
Taken reference from
( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )
The probability P(E) of happening of an event E satisfies the following properties :
<!-- [if !supportLists]-->· <!--[endif]-->p + q = 1, where q is the probability of not happening of the event.
<!-- [if !supportLists]-->· <!--[endif]-->If P(E) = 1 , it is an certain event.
<!-- [if !supportLists]-->· <!--[endif]-->If P(E) = 0 , it is an impossible event.
<!-- [if !supportLists]-->· <!--[endif]-->$$\0<P(E)<1$$
Two events are said to be dependent if the occurrence of one event in a trial affects the probability of the occurrence of the other event in other trial.
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