mathematics-class-12-unit-13-chapter-12-probability-l-2-6-yiheo1d1fby

### <Success style="font-size:25px"> Probability L-2 </Success>
### <primary style="font-size:24px"> Application </primary>
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- In how many ways you can choose 5 numbers $a, b, c, d, e$ there-exist

- each one is $>0$ \& $a+b+c+d+e=20$
- then the number of possible

- Solution: in $\quad{ }^{20-1} c_{5-1}={ }^{19} c_4$

- However, if $a, b, c, d, e \geq 0$ then the number of possible

- sol's is $^{20+5-1}C_{5-1}$ $={ }^{24} c_4$

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<footer style = "font-size: 18px">n-identical balls case-1 $\rightarrow$ n-identical balls case-2 $\rightarrow$ <span style = "color:blue"> Application </span> $\rightarrow$ Problem-1 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Probability L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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| $x_1$ | $x_2$ | $x_3$ | $x_4$ |
| -------- | -------- | -------- | -------- |
| 0    | 0     | 0   | 6   |
| 0    | 0    | 1  | 5   |
| 0    | 0     | 2 | 4    |
| 0    | 0     | 3 | 3    |
| 0    | 1     | 1   | 4   |
| 0    |1     | 2  | 3   |
| 1    | 1     | 1 | 3    |
| 1    | 1    | 2 | 2    |
| 0    | 2     | 2 | 2    |

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- 9 possible sol.
- $(x_1,x_2,x_3,x_4)$    

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<footer style = "font-size: 18px">Solution $\rightarrow$ Problem-3 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Probability $\rightarrow$ Event </footer>

### <Success style="font-size:25px"> Probability L-2 </Success>
### <primary style="font-size:24px"> Event </primary>
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- <ins>Event:</ins>     an event is a  subset of the sample space $\Omega$.

- <ins>consider :</ins> we toss a coin 3 time. we want to compute that number of H's is odd.
-  HTT THT TTH & HHH
 
- $\therefore$ We are looking at the probabilitiy of the subsect: $\{H T T, T H T, T T H, H H H\}$.

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<footer style = "font-size: 18px">Solution $\rightarrow$ Probability $\rightarrow$ <span style = "color:blue"> Event </span> $\rightarrow$ Elementary and compound events $\rightarrow$ Elementary and compound events </footer>

### <Success style="font-size:25px"> Probability L-2 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- Let A and B be two events $\exists$

- $0<P(A)<1$ \& $0<P(B)<1 \text {. }$

- Then which of the following statements are True?

- (A) $A$ and $A^c$ are mutually exclusive.

- (B) $A$ and $A^c$ are independent.

- (C) $A$ and $B$ are independent $\Rightarrow$ $A$ and $B^c$ are independent.

- (D) $A$ and $B$ are independent $\Rightarrow$ $A^c$ and $B^c$ are independent.

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<footer style = "font-size: 18px">Example-2 $\rightarrow$ Important property $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-3 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Probability L-2 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- (A) It is obvious that A & $A^c$ are mutually exclusive.

- $\because {\exists} \omega \geqslant \omega \in A  \omega \in A^c$.

- (B) $A$ and $A^c$ are independent Now we know that $A$ and $B$ are independent if

- $P(A \cap B)=P(A) P(B)$

- Now, $P\left(A \cap A^c\right)=0 $

- $ [\because \quad A \cap A^c$ in $\varnothing]$

- However, $P(A) * P\left(A^C\right) \neq 0 $

- $[ \therefore P\left(A \cap A^c\right)]\qquad
 \because O<P(A)<1  =\frac{ \phi}{ \Omega} =0$

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<footer style = "font-size: 18px">Important property $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-3 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Probability L-2 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- (D) $A$ and $B$ are independent $\Rightarrow$ $A^c$ and $B^c$ are independent

- $ P\left(A^c \cap B^c\right)  =P(\Omega)-P(A \cup B)$     
- $ =1-(P(A)+P(B)-P(A \cap B))$       
- $=1-P(A)-P(B)+P(A \cap B)$

- $ =(1-P(A))-P(B)(1-P(A))$

- $\because P(A \cap B)=P(A).P(B) $

- $=(1-P(A))(1-P(B)) $

- $=P\left(A^c\right) P\left(B^c\right)$

- $\therefore A^c  B^c$ are independent
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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>