Probability - Notes, Concept and All Important Formula

PROBABILITY

1. SOME BASIC TERMS AND CONCEPTS

(a) An Experiment: An action or operation resulting in two or more outcomes is called an experiment.

(b) Sample Space : The set of all possible outcomes of an experiment is called the sample space, denoted by S. An element of $S$ is called a sample point.

(c) Event : Any subset of sample space is an event.

(d) Simple Event : An event is called a simple event if it is a singleton subset of the sample space $S$.

(e) Compound Events : It is the joint occurrence of two or more simple events.

(f) Equally Likely Events : A number of simple events are said to be equally likely if there is no reason for one event to occur in preference to any other event.

(g) Exhaustive Events : All the possible outcomes taken together in which an experiment can result are said to be exhaustive.

(h) Mutually Exclusive or Disjoint Events : If two events cannot occur simultaneously, then they are mutually exclusive.

If $A$ and $B$ are mutually exclusive, then $A \cap B=\phi$.

(i) Complement of an Event : The complement of an event A, denoted by $\overline{A}$, $A'$  or $A ^{ C }$, is the set of all sample points of the space other then the sample points in $A$.

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All Chapter Notes, Concept and Important Formula

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2. MATHEMATICAL DEFINITION OF PROBABILITY

Let the outcomes of an experiment consists of $n$ exhaustive mutually exclusive and equally likely cases. Then the sample spaces $S$ has $n$ sample points. If an event A consists of $m$ sample points, $(0 \leq m \leq n)$, then the probability of event A, denoted by P(A) is defined to be $m/ n$ i.e. $P (A)= m / n$

Let $S=a_{1}, a_{2}, \ldots \ldots, a_{n}$ be the sample space

(a) $P ( S )=\frac{ n }{ n }=1$ corresponding to the certain event.

(b) $P (\phi)=\frac{0}{ n }=0$ corresponding to the null event $\phi$ or impossible event.

(c) If $A_{i}=\left\{a_{i}\right\}, i=1, \ldots . ., n$ then $A_{i}$ is the event corresponding to a single sample point $a_{i} .$ Then $P\left(A_{i}\right)=\frac{1}{n}$.

(d) $0 \leq P ( A ) \leq 1$

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3. ODDS AGAINST AND ODDS IN FAVOUR OF AN EVENT :

Let there be $m + n$ equally likely, mutually exclusive and exhaustive cases out of which an event A can occur in m cases and does not occur in $n$ cases. Then by definition, probability of occurrences of event $A=P(A)=\dfrac{m}{m+n}$

The probability of non-occurrence of event $A=P\left(A^{\prime}\right)=\dfrac{n}{m+n}$ 

$\therefore P ( A ): P \left( A ^{\prime}\right)= m : n$

Thus the odd in favour of occurrences of the event $A$ are defined by $m : n$ i.e. $P ( A ): P \left( A ^{\prime}\right) ;$ and the odds against the occurrence of the event $A$ are defined by $n:$ m i.e. $P\left(A^{\prime}\right): P(A)$.

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4. ADDITION THEOREM

(a) If $A$ and $B$ are any events in $S$, then $P ( A \cup B )= P ( A )+ P ( B )- P ( A \cap B )$

Since the probability of an event is a nonnegative number, it follows that

$P(A \cup B) \leq P(A)+P(B)$

For three events $A, B$ and $C$ in $S$ we have $P(A \cup B \cup C)$$=P(A) +P(B)+P(C)$$-P(A \cap B)-$$P(B \cap C)$$-P(C \cap A)+$$P(A \cap B \cap C)$

General form of addition theorem (Principle of Inclusion-Exclusion) 

For $n$ events $A _{1}, A _{2}, A _{3}, \ldots \ldots A _{ n }$ in $S$, we have

$P \left( A _{1} \cup A _{2} \cup A _{3} \cup A _{4} \ldots \ldots \ldots \cup A _{ n }\right)$

$=\displaystyle \sum_{i=1}^{n} P ( A_i)-$$\displaystyle \sum_{i<j} P \left( A _{i} \cap A _{j}\right)+$$\displaystyle\sum_{i<j<k} P \left( A _{i} \cap A _{j} \cap A _{k}\right)+\ldots$$+(-1)^{ n -1} P \left( A _{1} \cap A _{2} \cap A _{3} \ldots \ldots \cap A _{ n }\right)$

(b) If $A$ and $B$ are mutually exclusive, then $P(A \cap B)=0$ so that $P ( A \cup B )= P ( A )+ P ( B )$

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5. CONDITIONAL PROBABILITY :

If $A$ and $B$ are any events in $S$ then the conditional probability of B relative to A, i.e. probability of occurence of $B$ when A has occured, is given by

$P ( B / A )=\dfrac{ P ( B \cap A )}{ P ( A )}$ .  If $P ( A ) \neq 0$

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6. MULTIPLICATION THEOREM

Independent event:

So if $A$ and $B$ are two independent events then happening of $B$ will have no effect on $A$.

(a) When events are independent:

$P ( A / B )= P ( A )$ and $P ( B / A )= P ( B )$, then

$P(A \cap B)=P(A) . P(B)$ or $P(A B)=P(A) \cdot P(B)$

(b) When events are not independent 

The probability of simultaneous happening of two events $A$ and $B$ is equal to the probability of A multiplied by the conditional probability of $B$ with respect to A (or probability of B multiplied by the conditional probability of A with respect to B) i.e

$P(A \cap B)=P(A) \cdot P(B / A)$ or $P(B) \cdot P(A / B)$ OR

$P(A B)=P(A) \cdot P(B / A) \text { or } P(B) . P(A / B)$

(c) Probability of at least one of the n Independent events

If $p _{1}, p _{2}, p _{3}, \ldots \ldots p _{ n }$ are the probabilities of $n$ independent events $A _{1}, A _{2}, A _{3} \ldots \ldots A _{ n }$ then the probability of happening of at least one of these event is

$1-\left[\left(1- p _{1}\right)\left(1- p _{2}\right) \ldots \ldots\left(1- p _{ n }\right)\right]$

$\Rightarrow P \left( A _{1}+ A _{2}+ A _{3}+\ldots+ A _{ n }\right)$$=1- P \left(\overline{ A }_{1}\right) P \left(\overline{ A }_{2}\right) P \left(\overline{ A }_{3}\right) \ldots$$\ldots P \left(\overline{ A }_{ n }\right)$

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7. TOTAL PROBABILITY THEOREM :

Let an event $A$ of an experiment occurs with its $n$ mutually exclusive & exhaustive events $B _{1}, B _{2}, B _{3}, \ldots \ldots . B _{ n }$ then total probability of occurence of even A is

$P ( A )$$= P \left( AB _{1}\right)+ P \left( AB _{2}\right)+\ldots$$\ldots+ P \left( AB _{ n }\right)$$=\displaystyle \sum_{ i =1}^{ n } P \left( AB _{ i }\right)$

$\Rightarrow P ( A )$$= P \left( B _{1}\right) P \left( A \mid B _{1}\right)+ P \left( B _{2}\right) P \left( A \mid B _{2}\right)+\ldots$$\ldots+ P \left( B _{ n }\right) P \left( A \mid B _{ n }\right)$$=\sum P \left( B _{ i }\right) P \left( A / B _{ i }\right)$

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8. BAYE'S THEOREM OR INVERSE PBOBABILITY :

Let $A_{1}, A_{2,} \ldots . ., A_{n}$ be $n$ mutually exclusive and exhaustive events of the sample space $S$ and $A$ is event which can occur with any of the events then $P \left(\dfrac{ A _{ i }}{ A }\right)=\dfrac{ P \left( A _{ i }\right) P \left( A / A _{ i }\right)}{\displaystyle \sum_{ i =1}^{ n } P \left( A _{ i }\right) P \left( A / A _{ i }\right)}$

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9. BINOMIAL DISTRIBUTION FOR REPEATED TRIALS

Binomial Experiment : Any experiment which has only two outcomes is known as binomial experiment.

Outcomes of such an experiment are known as success and failure.

Probability of success is denoted by $p$ and probability of failure by $q$

$\therefore p + q =1$

If binomial experiment is repeated $n$ times, then

(a) Probability of exactly $r$ successes in $n$ trials $={ }^{ n } C _{ r } p ^{ r } q ^{ n - r }$

(b) Probability of at most $r$ successes in $n$ trails $=\displaystyle \sum_{\lambda=0}^{ r }{ }^{n} C_{\lambda} p^{\lambda} q^{n-\lambda}$

(c) Probability of atleast $r$ successes in $n$ trails $=\displaystyle \sum_{\lambda=r}^{n}{ }^{n} C_{\lambda} p^{\lambda} q^{n-\lambda}$

(d) Probability of having $I ^{ st }$ success at the $r ^{ th }$ trials $= p q ^{ r -1}$. The mean, the variance and the standard deviation of binomial distribution are $np , npq , \sqrt{ npq }$.

Note : $(p+q)^{n}=$${ }^{n} C_{0} q^{n}+{ }^{n} C_{1} p q^{n-1}+{ }^{n} C_{2} p^{2} q^{n-2}+\ldots$$...+{ }^{n} C_{r} p^{r} q^{n-r}+\ldots$$. \ldots+{ }^{n} C_{n} p^{n}=1$

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10. SOME IMPORTANT RESULTS

(a) Let $A$ and $B$ be two events, then

(i) $P(A)+P(\bar{A})=1$

(ii) $P ( A + B )=1- P (\bar{ A } \bar{ B })$

(iii) $P(A / B)=\dfrac{P(A B)}{P(B)}$

(iv) $P ( A + B )= P ( AB )+ P (\overline{ A } B )+ P ( A \overline{ B })$

(v) $A \subset B \Rightarrow P(A) \leq P(B)$

(vi) $P(\bar{A} B)=P(B)-P(A B)$

(vii) $P(A B) \leq P(A) P(B) \leq P(A+B) \leq P(A)+P(B)$

(viii) $P ( AB )= P ( A )+ P ( B )- P ( A + B )$

(ix) $P ($ Exactly one event $)= P ( A \overline{ B })+ P (\overline{ A } B )$

$= P ( A )+ P ( B )-2 P ( AB )= P ( A + B )- P ( AB )$

(x) $P$ (neither A nor $B)=P(\bar{A} \bar{B})=1-P(A+B)$

(xi) $P(\bar{A}+\bar{B})=1-P(A B)$

(b) Number of exhaustive cases of tossing $n$ coins simultaneously (or of tossing a coin $n$ times $)=2^{ n }$

(c) Number of exhaustive cases of throwing $n$ dice simultaneously (or throwing one dice $n$ times $)=6^{ n }$

(d) Playing Cards :

(i) Total Cards: 52(26 red, 26 black)

(ii) Four suits : Heart, Diamond, Spade, Club - 13 cards each

(iii) Court Cards : 12 (4 Kings, 4 queens, 4 jacks)

(iv) Honour Cards: 16 (4 aces, 4 kings, 4 queens, 4 jacks)

(e) Probability regarding n letters and their envelopes:

If n letters are placed into n directed envelopes at random, then

(i) Probability that all letters are in right envelopes $=\dfrac{1}{n !}$.

(ii) Probability that all letters are not in right envelopes $=1-\dfrac{1}{n !}$

(iii) Probability that no letters is in right envelopes

$=\dfrac{1}{2 !}-\dfrac{1}{3 !}+\dfrac{1}{4 !}-\ldots . .+(-1)^{ n } \dfrac{1}{ n !}$

(iv) Probability that exactly $r$ letters are in right envelopes $=\dfrac{1}{r !}\left[\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\ldots . .+(-1)^{n-r} \frac{1}{(n-r) !}\right]$

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11. PROBABILITY DISTRIBUTION :

(a) A Probability Distribution spells out how a total probability of 1 is distributed over several values of a random variable.

(b) Mean of any probability distribution of a random variable is given by:

$\mu=\frac{\sum p _{ i } x _{ i }}{\sum p _{ i }}=\sum p _{ i } x _{ i }\left(\right.$ Since $\left.\Sigma p _{ i }=1\right)$

(c) Variance of a random variable is given by, $\sigma^{2}=\sum\left( x _{ i }-\mu\right)^{2} \cdot p _{ i }$

$\sigma^{2}=\sum p _{ i } x _{ i }^{2}-\mu^{2}$ (Note that Standard Deviation $( SD )=+\sqrt{\sigma^{2}}$

(d) The probability distribution for a binomial variate ' $X$ ' is given  by $P  ( X = r )$$={ }^{ n } C _{ r }  p ^{ r }  q ^{ n - r }$ where: $p =$ probability of success in a single trial, $q =$ probability of failure in a single trial and $p + q =1$.

(e) Mean of Binomial Probability Distribution (BPD) $= np$; variance of $BPD = npq$

(f) If p represents a person's chance of success in any venture and 'M' the sum of money which he will receive in case of success, then his expectations or probable value $= pM$

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