mathematics-class-12-unit-13-chapter-02-probability-introduction-to-probability-lecture-2-10-eqlsce-ts6s

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Theory of probability </primary>
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- **Probability**

- The theory of probability originated in the middle of seventeenth century.

- Fermat (1601-1665), Pascal (1623-1662) Hugger (1629-1695) James Bernoulli: (1663)

- Experiment : An experiment is observing something happen or conducting something which results in an outcome.
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<footer style = "font-size: 18px"> $\rightarrow$ Probability lecture-2 $\rightarrow$ <span style = "color:blue"> Theory of probability </span> $\rightarrow$ Deterministic experiment $\rightarrow$ Sample space </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Deterministic experiment </primary>
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- Deterministic Expt $\rightarrow$ If after conducting the expt we know the outcome of the expt.

- $\mathrm{H}_2 \mathrm{O} \rightarrow \text { water }$

- Non-deterministic or Random Experiments When an expt is conducted but the outcome cannot be predicted in advance, then it is called a random expt.

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<footer style = "font-size: 18px">Probability lecture-2 $\rightarrow$ Theory of probability $\rightarrow$ <span style = "color:blue"> Deterministic experiment </span> $\rightarrow$ Sample space $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- 3. Time of the day

- S=(m, n, p)

- $m=1,2, \ldots, 12,$

- $n=0,1, \ldots, 54$

- $p=0,1, \ldots 59 .$

- If we consider continuous time, then we may write the sample space as

- $\Omega=(0,24)$

- 4. Suppose we consider a $100 \mathrm{nt}$ sprint race, say' of olympic standard

- $P_1, \ldots, p_8$

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<footer style = "font-size: 18px">Sample space $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Examples $\rightarrow$ Event </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- If we record the winner, then the sample space is $\quad S_1={P_1, \cdots, P_8}$

- If we are interested in the time $f$ the winner then the sample space can be

- $S_2=(9: 50,10: 00) \quad \text { (in seconds) }$

- 5. Suppose we record the number of accidents taking place in a city in an year.

- $S=\{0, 1,2, \ldots \ldots .\}$

- life of an organism:
- $
\begin{aligned}
& {[0,100)} \\
& {[0,150)}
\end{aligned}
 $

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<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Event $\rightarrow$ Example of an event </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Example of an event </primary>
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- Amount of rainfall during a monsoon season $S=[0,200)$ (in cm)

- $E=[50,75] \rightarrow$ the amount of rainfall is between $50$ to $75 \mathrm{~cm}$.
- $ E \subset S$

- Sure event: $\rightarrow$ the event that will certainly happen

- $\rightarrow$ We use $S$ to denote the sure event

- Impossible event:

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<footer style = "font-size: 18px">Examples $\rightarrow$ Event $\rightarrow$ <span style = "color:blue"> Example of an event </span> $\rightarrow$ Union of two events $\rightarrow$ Intersection of events </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Union of two events </primary>
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- Let $A$ and $B$ be tor events

- $A \cup B \rightarrow$ Occurrence of either $A$ or $B$ or (both occurence of atleast one of A and B)

- $ \bigcup_{i=1}^n A_i=A_1 \cup A_2 \cup \cdots \cup A_n $

- $ =\text { occurrence of at leas one } A_i $

- $(i=1, \cdots n)$

- $\bigcup_{i=1}^{\infty} A_i=$ occurrence of at least one $A_i$

- $(i=1,2, \ldots)$

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<footer style = "font-size: 18px">Event $\rightarrow$ Example of an event $\rightarrow$ <span style = "color:blue"> Union of two events </span> $\rightarrow$ Intersection of events $\rightarrow$ Disjoint or mutually exclusive events </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Intersection of events </primary>
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- $A \cap B \rightarrow$ simultaneous occurrence of both $A$ and $B$

- $\cap_{i=1}^n A_i=A_1 \cap A_2 \cap \cdots \cap A_n$

- $\rightarrow$ simultaneous occurrence of $A_1, \ldots, A_n$

- $\bigcap_{i=1}^{\infty} A_i$
- $ \rightarrow$ simultaneous occurrence of $A_1, A_2, \ldots$.

- $A-B=A \cap B^c$

- $\rightarrow$ Occurrence of A but not of B

- $\rightarrow$ A occur but B does not occur

- $A^c \rightarrow \text { not occurrence of } A$

- $A \cap B=\phi $ $\rightarrow$ disjoint or mutually exclusive events.
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<footer style = "font-size: 18px">Example of an event $\rightarrow$ Union of two events $\rightarrow$ <span style = "color:blue"> Intersection of events </span> $\rightarrow$ Disjoint or mutually exclusive events $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Probability Introduction To Probability L-2 </Success>
### <primary style="font-size:24px"> Disjoint or mutually exclusive events </primary>
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- Let $A_1, \ldots$. $A_n$ be any $\alpha$ events and $A_i \cap A_j=\phi$ for $i \neq j$.
- Then $A_1, A_2, \ldots A_n$ are called pairwise disjoint events .
- Exhaustive events: If $\bigcup_{i=1}^n A_i=S$, then we say that events $A_1, \ldots$. $A_n$ as  exhaustive events.

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<footer style = "font-size: 18px">Union of two events $\rightarrow$ Intersection of events $\rightarrow$ <span style = "color:blue"> Disjoint or mutually exclusive events </span> $\rightarrow$ Example $\rightarrow$ Thank you </footer>