Download App

Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability5 Marks
Question
Prove that:
  1. $\text{P}(\text{A})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})$
  2. $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})+\text{P}(\bar{\text{A}}\cap\text{B})$
✓

Answer

  1. $\because\text{P}(\text{A})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})$
$\therefore\text{R.H.S.}=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})$

$=\text{P}(\text{A})\cdot\text{P}(\text{B})+\text{P}(\text{A})\cdot\text{P}\bar{(\text{B})}$

$=\text{P}(\text{A})\big[\text{P}(\text{B})+\text{P}\bar{(\text{B})}\big]$

$=\text{P}(\text{A})\big[\text{P}(\text{B})+1-\text{P}(\text{B})\big]$ $\big[\because\text{P}\bar{(\text{B})}=1-\text{P}(\text{B})\big]$

$= \text{P(A) = L. H. S} $ Hence proved.
  1. $\because\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})+\text{P}(\bar{\text{A}}\cap\text{B})$
$\therefore\text{R.H.S.}=\text{P}(\text{A})\cdot\text{P}(\text{B})+\text{P}(\text{A})\cdot\text{P}\bar{(\text{B})}+\text{P}\bar{(\text{A})}\cdot\text{P}(\text{B})$

$=\text{P}(\text{A})\cdot\text{P}(\text{B})+\text{P}(\text{A})\cdot\big[1-\text{P}(\text{B})\big]+\big[1-\text{P}(\text{A})\big]\text{P}(\text{B})$

$=\text{P}(\text{A})\cdot\text{P}(\text{B})+\text{P}(\text{A})-\text{P}(\text{A})\cdot\text{P}(\text{B})+\text{P}(\text{B})-\text{P}(\text{A})\cdot\text{P}(\text{B})$

$=\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A})\cdot\text{P}(\text{B})$

$=\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})$

$=\text{P}(\text{A}\cup\text{B})=\text{L.H.S.}$ Hence proved.

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 "5 Marks Questions" from Probability→Probability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equation:
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$
→
Solve $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}(\log{\text{y}-\log\text{x}+1}).$
→
Maximize Z = 3x + 3y, if possible,
Subject to the constraints
$\text{x}-\text{y}\leq1$
$\text{x}+\text{y}\geq3$
$\text{x},\text{y}\geq0$
→
A(1, 0, 4) B(0, -11, 1), C(2, -3, 1) are three points and D is the fool of perpendicular from A on BC. Find the coordinates of D.
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=(\text{x}+1)(\text{x}+2)^\frac{1}{3}$
→
Solve the following differential equation
$\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}=\tan^{-1}\text{x},\text{x}\neq0$
→
Examine the differentiability of f, where f is defined by:
$\text{f(x)}=\begin{cases}\text{x}^2\sin\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$
at x = 0.
→
Evaluate the following integrals:
$\int\cos^3\sqrt{\text{x}}\text{dx}$
→
The contents of three bags I, II and III are as follows:
Bag I : 1 white, 2 black and 3 red balls,
Bag II : 2 white, 1 black and 1 red ball;
Bag III : 4 white, 5 black and 3 red balls.
A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red?
→
Differentiate $\sin^{-1}\Big\{\frac{2^{\text{x}+1}\times3^\text{x}}{1+(36)^\text{x}}\Big\}$ with respect to x:
→