Download App

13. probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths13. probability1 Mark
MCQ
Let $E ^{ C }$ denote the complement of an event $E$. Let $E _{1}, E _{2}$ and $E _{3}$ be any pairwise independent events with $P \left( E _{1}\right) > 0$ and $P \left( E _{1} \cap E _{2} \cap E _{3}\right)=0$ Then $P \left( E _{2}^{ C } \cap E _{3}^{ C } / E _{1}\right)$ is equal to
  • ✓
    $\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}\right)$
  • B
    $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)+\mathrm{P}\left(\mathrm{E}_{3}\right)$
  • C
    $\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)$
  • D
    $\mathrm{P}\left(\mathrm{E}_{3}\right)-\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)$
✓

Answer

Correct option: A.
$\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}\right)$
a
Given $\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$ are pairwise indepedent events $\operatorname{soP}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)$

and $\mathrm{P}\left(\mathrm{E}_{2} \cap \mathrm{E}_{3}\right)=\mathrm{P}\left(\mathrm{E}_{2}\right) \cdot \mathrm{P}\left(\mathrm{E}_{3}\right)$

and $\mathrm{P}\left(\mathrm{E}_{3} \cap \mathrm{E}_{1}\right)=\mathrm{P}\left(\mathrm{E}_{3}\right) \cdot \mathrm{P}\left(\mathrm{E}_{1}\right)$

$\& \mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2} \cap \mathrm{E}_{3}\right)=0$

Now $\mathrm{P}\left(\frac{\overline{\mathrm{E}}_{2} \cap \overline{\mathrm{E}}_{3}}{\mathrm{E}_{1}}\right)=\frac{\mathrm{P}\left[\mathrm{E}_{1} \cap\left(\overline{\mathrm{E}}_{2} \cap \overline{\mathrm{E}}_{3}\right)\right]}{\mathrm{P}\left(\mathrm{E}_{1}\right)}$

$=\frac{\mathrm{P}\left(\mathrm{E}_{1}\right)-\left[\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2}\right)+\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{3}\right)-\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2} \cap \mathrm{E}_{3}\right)\right]}{\mathrm{P}\left(\mathrm{E}_{1}\right)}$

$=\frac{\mathrm{P}\left(\mathrm{E}_{1}\right)-\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)-\mathrm{P}\left(\mathrm{E}_{1}\right) \mathrm{P}\left(\mathrm{E}_{3}\right)-0}{\mathrm{P}\left(\mathrm{E}_{1}\right)}$

$=1-\mathrm{P}\left(\mathrm{E}_{2}\right)-\mathrm{P}\left(\mathrm{E}_{3}\right)$

$=\left[1-\mathrm{P}\left(\mathrm{E}_{3}\right)\right]-\mathrm{P}\left(\mathrm{E}_{2}\right)$

$=\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{c}}\right)-\mathrm{P}\left(\mathrm{E}_{2}\right)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 13. probability→13. probability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate $\begin{bmatrix}\text{x}^2&\text{x}^3&\text{x}^4\\\text{x}&\text{y}&\text{z}\\\text{x}^2&\text{x}^3&\text{x}^4\end{bmatrix}$ is:
  1. c
  2. 1
  3. xyz
  4. x2 yz3
→
What is the maximum value of function $f(x)=\sin x+\cos x$ ?
→
If $A$ and $B$ are any two events such that $P ( A )+ P ( B )- P ( A$ and B $)= P ( A )$ then __________ .
→
Let $\text{f(x)}=\frac{1}{1-\text{x}}.$ Then, {fo(fof)}(x):
  1. x for all $\text{x}\in\text{R}$
  2. x for all $\text{x}\in\text{R}-\{1\}$
  3. x for all $\text{x}\in\text{R}-\{0,1\}$
  4. None of these.
→
If $\triangle_1=\begin{vmatrix}1&1&1\\\text{a}&\text{b}&\text{c}\\\text{a}^2&\text{b}^2&\text{c}^2\end{vmatrix},\triangle_2=\begin{vmatrix}1&\text{bc}&\text{a}\\1&\text{ca}&\text{b}\\1&\text{ab}&\text{c}\end{vmatrix},$ then:
  1. $\triangle_1+\triangle_2=0$
  2. $\triangle_1+2\triangle_2=0$
  3. $\triangle_1=\triangle_2$
  4. None of these.
→
Let $A$ be $2$$ \times $$2$ matrix

Statement $-1 :$  $adj\left( {adj\;A} \right) = A$

Statement $-2 :$ $\left| {adj\;A} \right| = \left| A \right|$

→
Choose the correct answer in each of the following:
If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1, then
→
The differential equation representing the family of curves ${y^2} = \sqrt c (x + 2c)$, where $c$ is $a$ positive parameter, is of 
→
Area bounded by curve $y = {x^3},$ $x - $ axis and ordinates $x = 1$ and $x = 4,$ is
→
The value of $\big[\vec{\text{a}}-\vec{\text{b}},\vec{\text{b}}-\vec{\text{c}},\vec{\text{c}}-\vec{\text{a}}\big],$ where $\big|\vec{\text{a}}\big|=1,\big|\vec{\text{b}}\big|=5,\big|\vec{\text{c}}\big|=3,$ is:
  1. 0
  2. 1
  3. 6
  4. None of these.
→