mathematics-class-12-unit-13-chapter-10-probability-definitions-of-probability-l-10-10-z1wg3x4axhg

### <Success style="font-size:25px"> Probability Definitions Of Probability L-10 </Success>
### <primary style="font-size:24px"> Classical or mathematical definition of probability </primary>
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- Classical or Mathematical Definition of Probability (Laplace): Suppose a random experiment has

- [N] possible outcomes which are equally likely, mutually exclusive, exhaustive. Let $E$ be an event so that $M$ of these outcomes are favourable to happening of the event $E$. Then we define the probability of event $E$ as
- $
 P(E)=\frac{M}{N} \text { }
 $

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<footer style = "font-size: 18px"> $\rightarrow$ Probability lecture-10 $\rightarrow$ <span style = "color:blue"> Classical or mathematical definition of probability </span> $\rightarrow$ Relative frequency or empirical definition of probability $\rightarrow$ Theoretical experiment </footer>

### <Success style="font-size:25px"> Probability Definitions Of Probability L-10 </Success>
### <primary style="font-size:24px"> Relative frequency or empirical definition of probability </primary>
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- Suppose we are considering life of an electronic equipment $(0,60)$
- Here $N$ is not finish.

- Relative Frequency or Empirical Def' of Probability<ins> Von Mises.</ins>

- Suppose a random exp is conducted a large number of times independently under identical conditions. Suppose in $n$ trials of the expt., an event $E$ occurs $a_n$ times. 
- then of the limit $\lim _{n \rightarrow \infty} \frac{a_n}{n}$ exists, we define.

- $
 P(E)=\lim _{n \rightarrow \infty}\left(\frac{a_n}{n}\right)
 $
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<footer style = "font-size: 18px">Probability lecture-10 $\rightarrow$ Classical or mathematical definition of probability $\rightarrow$ <span style = "color:blue"> Relative frequency or empirical definition of probability </span> $\rightarrow$ Theoretical experiment $\rightarrow$ Theoretical experiment </footer>

### <Success style="font-size:25px"> Probability Definitions Of Probability L-10 </Success>
### <primary style="font-size:24px"> Theoretical experiment </primary>
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- Let us consider one theoretical expt. (Hypothetical Suppose an hypothetical sequence of tosses of a coin results in the following outcomes:
- HHHT, HHHT, HHHT, ......

- We want to find the prob. of head
- $
\begin{aligned}
& \text { (PH) } \quad \frac{a}{A} \rightarrow \frac{1}{1}, \frac{2}{2}, \frac{3}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{6}{8}, \\
& \frac{7}{9}, \frac{8}{10}, \frac{9}{11}, \frac{9}{12}, \ldots . \\
&
\end{aligned}
 $                
- So we can express

- $ \frac{a_n}{n} \quad =\frac{3 k}{4 k}, \text { of } n=4 k$                  
- $ =\frac{3 k}{4 k-1},\quad  \text { of } n=4 k-1$               
- $ =\frac{3 k-1}{4 k-2}, \quad \text { of } n=4 k-2$                  
- $ =\frac{3 k-2}{4 k-3} .\quad \text { of } n=4 k-3 $ $\qquad$ $[\because k=1,2,........]$

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<footer style = "font-size: 18px">Classical or mathematical definition of probability $\rightarrow$ Relative frequency or empirical definition of probability $\rightarrow$ <span style = "color:blue"> Theoretical experiment </span> $\rightarrow$ Theoretical experiment $\rightarrow$ Drawbacks </footer>

### <Success style="font-size:25px"> Probability Definitions Of Probability L-10 </Success>
### <primary style="font-size:24px"> Drawbacks </primary>
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-  1. We should have sufficient number of trials and their outcomes recorded.

- 2. Suppose  $a_n=\sqrt{n}, \quad \frac{a_n}{n}=\frac{\sqrt{n}}{n} \rightarrow 0$
- $
 4 \rightarrow 2, \quad 9 \rightarrow 3, \quad 16 \rightarrow 4
 $

- Probability of rare events is zero, but it does it mean that the event does not occur.

- 3. Let $a_n=n-n^{1 / 3}$
- $
\frac{a_n}{n}=\frac{n-n^{1 / 3}}{n}=1-\frac{1}{n^{2 / 3}} \rightarrow 1 \text {. }
 $

- Probability of events which may not occur sometimes is also 1.
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<footer style = "font-size: 18px">Theoretical experiment $\rightarrow$ Theoretical experiment $\rightarrow$ <span style = "color:blue"> Drawbacks </span> $\rightarrow$ Sample space conditions $\rightarrow$ Three axioms of probability </footer>

### <Success style="font-size:25px"> Probability Definitions Of Probability L-10 </Success>
### <primary style="font-size:24px"> Sample space conditions </primary>
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- Suppose $S$ be a sample space.

- Let us consider a class (set) of subsets of $S$ Let us denote it by $\cancel{\zeta}$

- Let this class satisfy the following two conditions

- (i) If $ \quad E \in \cancel{\zeta} \Rightarrow E^c \in\cancel{\zeta}$.

- (ii) If $ \quad E_1, E_2, \ldots \in \cancel{\zeta} \Rightarrow \bigcup_{i=1}^{\infty} E_i \in \cancel{\zeta}$

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<footer style = "font-size: 18px">Theoretical experiment $\rightarrow$ Drawbacks $\rightarrow$ <span style = "color:blue"> Sample space conditions </span> $\rightarrow$ Three axioms of probability $\rightarrow$ Some consequences of the axiomatic definition </footer>

### <Success style="font-size:25px"> Probability Definitions Of Probability L-10 </Success>
### <primary style="font-size:24px"> Some consequences of the axiomatic definition </primary>
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- 1. $P(\phi)=0$

- In axiom 3, let us take $E_1=S, E_2=E_3=\cdots$ $=\phi$

- Then,

- $ P(S)=P(S)+P(\phi)+P(\phi)+\ldots$

- $ \Rightarrow \quad 1  =1+P(\phi)+P(\phi)+\ldots$

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<footer style = "font-size: 18px">Sample space conditions $\rightarrow$ Three axioms of probability $\rightarrow$ <span style = "color:blue"> Some consequences of the axiomatic definition </span> $\rightarrow$ Some consequences of the axiomatic definition $\rightarrow$ Some consequences of the axiomatic definition </footer>