### Mathematics Class 10 Lab Manual | 21 Lab Activities

Mathematics Lab Manual Class X   lab activities for class 10 with complete observation Tables strictly according to the CBSE syllabus also very useful & helpful for the students and teachers.

### PROBABILITY Part-2

PROBABILITY PART-2
This part is the continuation of Probability part-1. This Post is specially for the students after 10th standard. For complete knowledge first students study Probability Part-1 and then part-2
 Main points to be discussed here 1 Random Experiment 2 Different types of events 3 Variable description of the events 4 Conditional Probability 5 Multiplication Theorem on Probability 6 Laws of Total Probability 7 Baye's Theorem 8 Bernoulli's Trial 9 Mean and Variance of the Distribution 10 Shortcut Method of finding Mean, Variance and Standard Deviation.

Random Experiment:-
An experiment whose outcomes cannot be predicted or determined in advance are called a random experiment.

Elementary events:-
If a random experiment performed, then each of its outcome is known as an elementary event.

Sample space:-
The set of all possible outcomes of a random experiment is called the sample space.

Event:- A subset of the sample space is called an event.

Simple Event :-
An event having only one sample point is called the simple event. eg:- HH, TT, HHH, 222 .... etc.
Compound event:-
An event having more than one sample point is called a compound event. eg:- HT, HTH,  123 ... etc.

Mutually exclusive events:-
Two or more events are said to be mutually exclusive events if their intersection is a null set eg:- A and B are said to be mutually exclusive events If A∩B = Î¦

Exhaustive events:-
Two or more events are said to be exhaustive events, if their union is a sample space, For Example:-  A ∪ B ∪ C ∪ ..... = S

Conditional probability:-
Conditional probability of two events A and B is denoted by P(A|B) and is called as
P(A|B) = Probability of occurrence  of A given that B has already occurred.

Conditional Probability is calculated by using following formula.
 $P(A|B)=\frac{P(A\cap B)}{P(B)}$

Independent events:- If A and B are independent events then
 $P(A\cap B)=P(A).P(B)$ $P(A\cap B\cap C)=P(A).P(B).P(C)$

Explanation of Some symbols and Terms used in the problems
 Variable Description of the event ⇒ Set theory notation A or B (At least one of A or B) = A ∪ B A and B = A ∩ B ,                    Not A = Ä€ $A \: but \: not\: B = A\cap\overline{B}$ $Either\; A\: or\: B=A\cup B$ $Neither\: A\: nor\: B=\overline{A}\cap \overline{B}$ $All \: three\: of\: A, B \: and \: C = A\cap B\cap C$ $At\: least\: one\: of\: A, B, or\: C=A\cup B\cup C$ $Exactly \: \: one\: of\: A\: and\: B = (A\cap \overline{B})\cup (\overline{A}\cap B)$ $D' Morgan's Law:-\: A'\cap B'=(A\cup B)'$$OR\: \: \: A'\cup B'=(A\cap B)'$ $P(A\cup B)=P(A)+P(B)-P(A\cap B)$$P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)$$-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)$
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 Multiplication Theorem On Probability If A and B are two events associated with a random experiment, then $P(A\cap B)=P(A)P(B/A)\: where P(A)\neq 0$ $P(A\cap B)=P(B)P(A/B)\: where P(B)\neq 0$ Extension of Multiplication Theorem $If A_{1},A_{2},A_{3},.....,A_{n}\: are\: n \: events\: associated\: with\: the\: random\: experiment,\: then$ $P(A_{1}\cap A_{2}\cap A_{3}.....\cap A_{n})=P(A_{1})P(A_{2}/A_{1})P(A_{3}/A_{1}\cap A_{2})P(A_{4}/A_{1}\cap A_{2}\cap A_{3}).....$ $........P(A_{n}/A_{1}\cap A_{2}\cap .....\cap A_{n-1})$
The laws of total probability:-
$Let\: S\: \: be\: the\: sample\: space \: and \: let\: E_{1},E_{2},E_{3},....,E_{n} \: be \: n \: mutually\: exclusive$$and\: exhaustive\: events.$$If \: A \: is \: any\: event,\: then$$P(A)=P(E_{1})P(A/E_{1})+P(E_{2})P(A/E_{2})+.....+P(E_{n})P(A/E_{n})$
 BAYE'S THEOREM $Let \: S\: be\: the \: sample\: space \: and\: let\: E_{1},E_{2},E_{3},.....,E_{n}\: be$ $mutually \: exclusive\: and\: exhaustive\: events\: associated\: with\: a\: random\: experiment.$ $If\: A \: is\: any \: event\: which\: occurs\: with\: E_{1} \: or\: E_{2}\: or\: E_{3}\: or...or\: E_{n}$ $P(E_{i}|A)=\frac{P(E_{i})P(A|E_{i})}{\sum P(E_{i})P(A|E_{i})}\: \: i=1,2,3....,n$If n = 2 then Baye's Theorem written as: $P(E_{1}|A) =\frac{P(E_{1})P(A|E_{1})}{P(E_{1})P(A|E_{1}) +P(E_{2}) P(A|E_{2})}$$P(E_{2}|A)=\frac{P(E_{2})P(A|E_{2})}{P(E_{1})P(A|E_{1})+ P(E_{2}) P(A| E_{2})}$If n = 3 then Baye's Theorem written as:$P(E_{1}|A)=\frac{P(E_{1})P(A|E_{1})}{P(E_{1})P(A|E_{1})+P(E_{2})P(A|E_{2}) +P(E_{3})P(A|E_{3})}$$P(E_{2}|A)=\frac{P(E_{2})P(A|E_{2})}{P(E_{1})P(A|E_{1})+P(E_{2})P(A|E_{2}) +P(E_{3})P(A|E_{3})}$ $P(E_{3}|A)=\frac{P(E_{3})P(A|E_{3})}{P(E_{1})P(A|E_{1})+P(E_{2})P(A|E_{2}) +P(E_{3})P(A|E_{3})}$
MEAN AND VARIANCE OF THE DISTRIBUTION
 $Mean(\overline{X})=\sum p_{i}x_{i}\: \: \: where\: \: i=1\: to\: n$$Variance(X) = \sum p_{i}x_{i}^{2}\: -\: \left ( \sum p_{i}x_{i} \right )^{2}\: \: \: where\: \: i=1\: to\: n$

BERNOULLI TRIAL
A trial of a random experiment are called Bernoulli trial, if they satisfy the following conditions:-

a) They are finite in number.

b) They are independent of each other.

c) Each trial has exactly two outcome: success or failure.

d) The probability of success or failure remain same in each trial.

BINOMIAL DISTRIBUTION
A random variable X which takes values 0,1,2,........,n is said to follow binomial distribution
if its probability distribution function is given by
 $P(X=r)=\:^{n}C_{r}\: p^{r}q^{n-r}, where r=1,2,3....n$
Where p and q are the probability of the two events such that   p + q = 1

n is the number of trials and r is the random variable .

The two constants n and p are called parameters of the distribution
$P(X=0)+P(X=1)+.....+P(X=n)=^{n}C_{0}p^{0}q^{n-0}+^{n}C_{1}p^{1}q^{n-1}.....^{n}C_{n}p^{n}q^{0}$
$\: \: \: \: \: \: =(p+q)^{n}=1^{n}=1$
 Bernoulli’s Trial Probability of r successes in ‘n’ Bernoulli trial is given by $P(r\: successes)=^{n}C_{r}p^{r}q^{n-r}$ $P(r\:successes)=\frac{n!}{r!(n-r)!}.p^{r}q^{n-r}$ Where  n = number of trials r = number of successful trials = 0, 1, 2, 3, ………., n p = probability of success in a trial q = probability of failure in a trial and  p + q = 1

The probability distribution of number of successes for a random variable X can be written as:

This probability distribution is called Binomial distribution with parameters n and p.

 Shortcut Method of finding Mean, Variance and Standard Deviation The binomial distribution with n Bernoulli trials and success p is also denoted by B(n, p)Where ‘n’ number of Bernoulli trial  p denotes the probability of success  q denotes the probability of failure then, Mean =  np Variance = npq Standard Deviation = Square root of  npq

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