Download App

STD 12 - 13. probability — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 13. probability4 Marks
MCQ
Two numbers are selected at random from the numbers $1, 2, ...... n.$ The probability that the difference between the first and second is not less than m (where $ o < m < n $ ) is
  • A
    $\frac{{(n - m)\,(n - m + 1)}}{{(n - 1)}}$
  • B
    $\frac{{(n - m)\,(n - m + 1)}}{{2n}}$
  • C
    $\frac{{(n - m)\,(n - m - 1)}}{{2n\,(n - 1)}}$
  • ✓
    $\frac{{(n - m)\,(n - m + 1)}}{{2n\,(n - 1)}}$
✓

Answer

Correct option: D.
$\frac{{(n - m)\,(n - m + 1)}}{{2n\,(n - 1)}}$
d
(d) Let the first number be $x$ and second is $y.$

Let $A$ denotes the event that the difference between the first and second number is at least $m.$

Let ${E_x}$ denote the event that the first number chosen is $x,$

we must have $x - y \ge m$ or $y \le x - m.$

Therefore $x > m$ and $y < n - m.$

Thus $P({E_x}) = 0$ for $0 < x \le m$ and $P({E_x}) = \frac{1}{n}$ for $m < x \le n.$

Also $P(A/{E_x}) = \frac{{(x - m)}}{{(n - 1)}}$

Therefore, $P(A) = \sum\limits_{x = 1}^n {P({E_x})\,\,P(A/{E_x})} $

$ = \sum\limits_{x = m + 1}^n {P({E_x})\,\,P(A/{E_x})} = \sum\limits_{x = m + 1}^n {\frac{1}{n}.\frac{{x - m}}{{n - 1}}} $

$ = \frac{1}{{n(n - 1)}}[1 + 2 + 3 + ..... + (n - m)]$

$ = \frac{{(n - m)\,(n - m + 1)}}{{2n(n - 1)}}.$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let ${S_k} = \frac{{1 + 2 + 3 + .... + k}}{k}$. If $S_1^2 + S_2^2 + ....... + S_{10}^2 = \frac{5}{{12}}A$, then $A$ equal to
→
The volume of spherical cap of height $h$ cut off from a sphere of radius $a$ is equal to
→
For a sequence $ < {a_n} > ,\;{a_1} = 2$ and $\frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{1}{3}$. Then $\sum\limits_{r = 1}^{20} {{a_r}} $ is
→
Let $ A$ be a $2$$ \times $$2$ matrix with non-zero entries and let ${A^2} = I$ where $I$  is $2\times 2$ identity matrix. Define $tr(A) =$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$ 

Statement $-1 :$ ${\rm{tr}}\left( A \right) = 0$

Statement $-2 :$  $\det \left( A \right) = 1$

→
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A ^{4}$ is equal to ............
→
Let $A$ and $B$ be two distinct points on the parabola $y^2=4 x$. If the axis of the parabola touches a circle of radius $r$ having $\mathrm{AB}$ as its diameter, then the slope of the line joining $\mathrm{A}$ and $\mathrm{B}$ can be

$(A)$ $-\frac{1}{\mathrm{r}}$ $(B)$ $\frac{1}{\mathrm{r}}$ $(C)$ $\frac{2}{\mathrm{r}}$ $(D)$ $-\frac{2}{r}$

→
Choose the correct statement about two circles whose equations are given below

$x^{2}+y^{2}-10 x-10 y+41=0$

$x^{2}+y^{2}-22 x-10 y+137=0$

→
If ${\rm{cosec}}\theta = \frac{{p + q}}{{p - q}},$ then $\cot \,\left( {\frac{\pi }{4} + \frac{\theta }{2}} \right) = $
→
The direction cosines of the normal to the plane $2x + 3y - 6z = 5$ are
→
If $A$ and $B$ are the coefficients of ${x^n}$ in the expansions of ${(1 + x)^{2n}}$ and ${(1 + x)^{2n - 1}}$ respectively, then
→