Class 11: Probability – Lecture Notes

$\displaystyle \textbf{1. Random Experiment}$
$\displaystyle \text{An experiment whose outcomes cannot be predicted or determined in advance is called a}$
$\displaystyle \text{random experiment.}$
$\\$

$\displaystyle \textbf{2. Elementary Event}$
$\displaystyle \text{Each outcome of a random experiment is known as an elementary event.}$
$\\$

$\displaystyle \textbf{3. Sample Space}$
$\displaystyle \text{The set of all possible outcomes (elementary events) of a random experiment is called the}$
$\displaystyle \text{sample space associated with it.}$
$\\$

$\displaystyle \textbf{4. Event}$
$\displaystyle \text{A subset of the sample space associated with a random experiment is called an event.}$
$\\$

$\displaystyle \textbf{5. Occurrence of an Event}$
$\displaystyle \text{An event is said to occur if any one of the elementary events belonging to it is an outcome.}$
$\\$

$\displaystyle \textbf{6. Certain Event}$
$\displaystyle \text{An event associated with a random experiment is called a certain event if it always occurs}$
$\displaystyle \text{whenever the experiment is performed.}$
$\displaystyle \text{The sample space associated with a random experiment defines a certain event.}$
$\\$

$\displaystyle \textbf{7. Impossible Event}$
$\displaystyle \text{The null set of the sample space defines an impossible event.}$
$\\$

$\displaystyle \textbf{8. Compound Event}$
$\displaystyle \text{An event associated with a random experiment is a compound event, if it is the disjoint}$
$\displaystyle \text{union of two or more elementary events.}$
$\\$

$\displaystyle \textbf{9. Mutually Exclusive Events}$
$\displaystyle \text{Two or more events associated with a random experiment are said to be mutually exclusive}$
$\displaystyle \text{or incompatible events if the occurrence of any one of them prevents the occurrence of all}$
$\displaystyle \text{others, i.e. no two or more of them can occur simultaneously in the same trial.}$
$\displaystyle \text{If }A\text{ and }B\text{ are mutually exclusive events, then }A\cap B=\phi.$
$\\$

$\displaystyle \textbf{10. Exhaustive Events}$
$\displaystyle \text{Events }A_1,A_2,A_3,\ldots,A_n\text{ associated with a random experiment with sample space }S$
$\displaystyle \text{are exhaustive if }A_1\cup A_2\cup\cdots\cup A_n=S.$
$\\$

$\displaystyle \textbf{11. Mutually Exclusive and Exhaustive System of Events}$
$\displaystyle \text{Let }S\text{ be the sample space associated with a random experiment. A set of events }$
$\displaystyle A_1,A_2,\ldots,A_n\text{ is said to form a set of mutually exclusive and exhaustive system}$
$\displaystyle \text{of events if}$
$\displaystyle \text{(i) }A_1\cup A_2\cup\cdots\cup A_n=S$
$\displaystyle \text{(ii) }A_i\cap A_j=\phi,\quad i\neq j$
$\\$

$\displaystyle \textbf{12. Probability Function}$
$\displaystyle \text{Let }S=\{w_1,w_2,\ldots,w_n\}\text{ be the sample space associated with a random experiment.}$
$\displaystyle \text{Then a function }P\text{ which assigns every event }A\subset S\text{ to a unique non-negative}$
$\displaystyle \text{real number }P(A)\text{ is called the probability function if the following axioms hold:}$
$\displaystyle \text{A-1: }0\leq P(w_i)\leq1\text{ for all }w_i\in S$
$\displaystyle \text{A-2: }P(S)=1,\text{ i.e. }P(w_1)+P(w_2)+\cdots+P(w_n)=1$
$\displaystyle \text{A-3: For any event }A\subset S,\;P(A)=\sum_{w_k\in A}P(w_k)$
$\displaystyle \text{The number }P(w_k)\text{ is called probability of elementary event }w_k.$
$\\$

$\displaystyle \textbf{13. Probability of an Event}$
$\displaystyle \text{If there are }n\text{ elementary events associated with a random experiment and }m\text{ of them}$
$\displaystyle \text{are favourable to an event }A,\text{ then the probability of occurrence of }A\text{ is defined as}$
$\displaystyle P(A)=\frac{m}{n}=\frac{\text{Favourable number of elementary events}}{\text{Total number of elementary events}}$
$\displaystyle \text{The odds in favour of occurrence of the event }A\text{ are defined by }m:(n-m).$
$\displaystyle \text{The odds against the occurrence of }A\text{ are defined by }(n-m):m.$
$\displaystyle \text{The probability of non-occurrence of }A\text{ is given by }P(\overline{A})=1-P(A).$
$\\$

$\displaystyle \textbf{14. Probability of Union of Two Events}$
$\displaystyle \text{If }A\text{ and }B\text{ are two events associated with a random experiment, then}$
$\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)$
$\displaystyle \text{If }A\text{ and }B\text{ are mutually exclusive events, then}$
$\displaystyle P(A\cup B)=P(A)+P(B)$
$\\$

$\displaystyle \textbf{15. Probability of Union of Three Events}$
$\displaystyle \text{If }A,B,C\text{ are three events associated with a random experiment, then}$
$\displaystyle P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)$
$\\$

$\displaystyle \textbf{16. Probability Relations for Two Events}$
$\displaystyle \text{If }A\text{ and }B\text{ are two events associated with a random experiment, then}$
$\displaystyle \text{(i) }P(\overline{A}\cap B)=P(B)-P(A\cap B)$
$\displaystyle \text{i.e. probability of occurrence of }B\text{ only }=P(B)-P(A\cap B)$
$\displaystyle \text{(ii) }P(A\cap\overline{B})=P(A)-P(A\cap B)$
$\displaystyle \text{i.e. probability of occurrence of }A\text{ only }=P(A)-P(A\cap B)$
$\displaystyle \text{(iii) Probability of occurrence of exactly one of }A\text{ and }B\text{ is}$
$\displaystyle P(A)+P(B)-2P(A\cap B)$
$\displaystyle =P(A\cup B)-P(A\cap B)$
$\\$