mathematics-class-12-unit-13-chapter-03-probability-some-rules-of-probability-l-3-10-qcdw5bgzhq

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> Axioms of probability </primary>
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- The following three axioms

- 1. P(E) $\geq$ 0 for all E $\in$ C 
- 2. P(S) = 1
- 3. let E_1, E_2,.... be pairwise disjoint  events then

- $P(\bigcup_{i=1}^{\infty} E_i)=\sum_{i=1}^{\infty} P\(E_i)$

- a,b,c are disjoint events

- $ P(A \cup B)=P(A)+P(B) $

- $ P(A \cup B \cup C)=P(A)+P(B)+P(C)$

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<footer style = "font-size: 18px"> $\rightarrow$ Probability lecture-3 $\rightarrow$ <span style = "color:blue"> Axioms of probability </span> $\rightarrow$ Addition rule of probability $\rightarrow$ Probability a monotonic function </footer>

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> $P(E^c)$ </primary>
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- 3)  $E \cup E^c=S $

- $P(E)+P\left(E^c\right)=P(S)=1 $

- $\Rightarrow P\left(E^c\right)=1-P(E) $

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<footer style = "font-size: 18px">Addition rule of probability $\rightarrow$ Probability a monotonic function $\rightarrow$ <span style = "color:blue"> $P(E^c)$ </span> $\rightarrow$ Addition rule of probability extension to 3 events $\rightarrow$ General addition rule </footer>

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> Addition rule of probability extension to 3 events </primary>
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- Extension to 3 events: $A, B, C$

- $ P(A \cup B \cup C)=P(A \cup B)+P(C)-P((A \cup B) \cap C) $

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

- $ \quad-P((\underbrace{A \cap C)} \cup(B \cap C)) $

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

- $ \quad-\quad(P(A \cap C)+P(B \cap C)-P((A \cap C) \cap(B \cap C))) $

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

- $ \quad-P(B \cap C)+P(A \cap B \cap C)$

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<footer style = "font-size: 18px">Probability a monotonic function $\rightarrow$ $P(E^c)$ $\rightarrow$ <span style = "color:blue"> Addition rule of probability extension to 3 events </span> $\rightarrow$ General addition rule $\rightarrow$ General addition rule proof </footer>

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> General addition rule </primary>
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- Let $A_1, A_2, \ldots, A_n$ be any events. Then

- $P(\bigcup_{i=1}^n A_i)=\sum_{i=1}^n P(A_i)-\underset{i<j}{\sum \sum} P(A_i \cap A_j)$

- $+ (\underset{i<j<k}){\sum\sum \sum}_k P(A_i \cap A_j \cap A_k)-\cdots $

- $+(-1)^{n+1} P(\bigcap_{i=1}^n A_i)$

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<footer style = "font-size: 18px">$P(E^c)$ $\rightarrow$ Addition rule of probability extension to 3 events $\rightarrow$ <span style = "color:blue"> General addition rule </span> $\rightarrow$ General addition rule proof $\rightarrow$ General addition rule proof </footer>

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> General addition rule proof </primary>
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- Proof: We will prove the relation (1) using the principle of mathematical induction.

- P(1)

- $P(K) \rightarrow P(K+1)$

- For $n=1$, the statement (1) becomes

- $P(A_1)=P(A_1)$ which always holds true.

- Next we assume the statement (1) to be true for $n=k$. Then we prove it for $n=k+1$

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<footer style = "font-size: 18px">Addition rule of probability extension to 3 events $\rightarrow$ General addition rule $\rightarrow$ <span style = "color:blue"> General addition rule proof </span> $\rightarrow$ General addition rule proof $\rightarrow$ General addition rule proof </footer>

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> General addition rule proof </primary>
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- $=  \sum_{i=1}^{k+1} P(A_i)- \underset{i<j}{\sum ^{k+1} \sum ^{k+1}} P (A_i \cap A_j) $

- $ + \underset{i<j<m} {\sum ^{k+1} \sum ^{k+1} \sum ^{k+1}} P (A_i \cap A_j \cap A_m) -\cdots $
- $ +(-1)^{k+2} P(\bigcap_{i=1}^{k+2} A_i )$

- this shows that statement (1) is true for $n=k+1$.

- Hence by the Principle of mathematical Induction the General addition rule holds for all $n$ ( $n$ is a positive integer).

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<footer style = "font-size: 18px">General addition rule proof $\rightarrow$ General addition rule proof $\rightarrow$ <span style = "color:blue"> General addition rule proof </span> $\rightarrow$ Problem $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Probability Some Rules Of Probability L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ P(B_i \cap B_j \cap B_k) = (\frac{1}{4})^6, \quad (1 \leq i<j<k \leq 4)$ 4 terms

- $ P(\bigcap_{i=1}^4 B_i)=0 $

- So, 
- $P(A^c) = 4 \times (\frac{3}{4})^6 - 6 \times (\frac{1}{2})^6 + 4 (\frac{1}{4})^6 = \frac{317}{512} \approx 0.62$

- $P(A) = 1 -P(A) = \frac{195}{512} \approx 0.38$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Thank you $\rightarrow$  </footer>