mathematics-class-12-unit-13-chapter-11-probability-l-1-6-hwotfrgth98

### <Success style="font-size:25px"> Probability L-1 </Success>
### <primary style="font-size:24px"> Random Experiments </primary>
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- $\rightarrow$It can be carried out any number of times under the identical condition.
- In these experiments the possible set of outcomes in known.

- But the outcome of any individual trial is unknown till it is execute.

- Ex Tossing a coin : $\{T, H\}$

- Ex Throwing a die : $\{1,2,3,4,5,6\}$

- Ex Tossing a coin three times \& noting the sequence of outcomes.

- {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

- Notation: $\Omega$ or $S$.
- So,  this is called the Sample space.
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<footer style = "font-size: 18px"> $\rightarrow$ Probability lecture-1 $\rightarrow$ <span style = "color:blue"> Random Experiments </span> $\rightarrow$ Example-1 $\rightarrow$ Example-2 </footer>

### <Success style="font-size:25px"> Probability L-1 </Success>
### <primary style="font-size:24px"> Example-2 </primary>
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- The random experiment in the following:                     
- suppose in as bus stop there  are four passengers. Each passengers may  have 1,2 or no bags. You come to the bus stop and pick up 1 or 2 or 3 or 4 passengers, and each passenger takes his/her bag?
- Describe the sample space on a pair of numbers (A  if passengers, \# of bags).

- Describe the sample space.                               
- (1,0),(1,1),(1,2), 
- (2,0),(2,1),(2,2),(2,3),(2,4)
- (3,0),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
- (4,0),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(4,7),(4,8)

- $\therefore|\Omega|=3+5+7+9=\underline{24} $

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<footer style = "font-size: 18px">Random Experiments $\rightarrow$ Example-1 $\rightarrow$ <span style = "color:blue"> Example-2 </span> $\rightarrow$ Infinite Sample Space $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Probability L-1 </Success>
### <primary style="font-size:24px"> Infinite Sample Space </primary>
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- Ex) You toss a coin repeatedly until you get H count the number of tosses required to get the first H. Describe the sample space
- number of tosses: $123 \ldots n$
- H TH TH
- $n$ can be any positive integer.
- $\therefore \Omega=\{1,2,3, \cdots\} $
- infinite

- Experiment with uncountable sample space

- Ex): Suppose we pick a point from the interval $(0,1)$
- The no. of possible ways \& selection is infinite. Point more importantly it in not countable.
- An infinite set $\Omega$ in said to be countable if there $n$ a 1 to l mapping between the elements of $\Omega$ and the set of natural numbers.

- If such a mapping doses not exist then it is uncountable.
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<footer style = "font-size: 18px">Example-1 $\rightarrow$ Example-2 $\rightarrow$ <span style = "color:blue"> Infinite Sample Space </span> $\rightarrow$ Example-3 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Probability L-1 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- Let us consider the following random experiment. Suppose we have 10 identical red balls we need to put them in 3 boxes s.t none 7 the box is empty. How many possible ways of putting the ball in the boxes?         
- $\rightarrow$ The balls are identical            
- $\rightarrow$ None of the boxes will be empty            
- Example:- Suppose we have 3 balls $A, B, C$ In how many ways we can put in 2 boxes $\rightarrow$ none is empty.            
- $
\begin{array}{ll}
 (A, B C) & (B, A C),(C, A B) \\
 (A B, C) & (A C, B)(B C, A)
\end{array}
 $         
- $\therefore 6$ possible ways

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

### <Success style="font-size:25px"> Probability L-1 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- $\therefore$ Total no of possibilities

- $ 1+2+\cdots+8=\frac{8 \times 9}{2}$

- $ =36 \text { different ways} $

* Suppose we have 100 identical balls to be put in 20 different boxes such that none of them is empty

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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-3 $\rightarrow$ Example-4 </footer>

### <Success style="font-size:25px"> Probability L-1 </Success>
### <primary style="font-size:24px"> Example-4 </primary>
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- In how many ways 3 positive numbers $x, y, z$ can be chosen a+ $x+y+z=15$
- Ans:- $\quad 1 \quad 1 \mid \begin{array}{llllllllllllll}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\end{array}$ Possible ways in ${ }^{14} c_2 \quad(2,8,5)$

- Consider the following :<br>
- We have 10 identical balls to be put in 3 boxes s.t any box can remain empty.
- What in the no of possibilities?

- $n+k$ elemfents can be put in $k$ boxes 3 none in empty is ${}^{n+k-1} c_{k-1}$

- Now we will have $n$ balls in $k$ boxes but some box can be empty also

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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-4 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>