Download App

Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability5 Marks
Question
A random variable $X$ has the following probability distribution:
$X$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$P(X)$ $0$ $k$ $2k$ $2k$ $3k$ $k^2$ $2k^2$ $7k^2+k$
Determine
  1. $k$
  2. $P(X < 3)$
  3. $P(X > 6)$
  4. $P(0 < X < 3)$
✓

Answer

Since, the sum of all the probabilities of a distribution is $1$.
$\therefore P(X = 0) + P(X = 1) + …. + P(X = 7) = 1$
$\Rightarrow 0 + k + 2k + 2k + 3k + k^2 + 2k^2+ 7k^2+ k = 1$
$\Rightarrow 10k^2 + 9k - 1 = 0$
$\Rightarrow (10k - 1) (k + 1) = 0$
$\Rightarrow 10k - 1 = 0$ or $k + 1 = 0$
$\Rightarrow\ \text{k}=\frac{1}{10}$ or $k = - 1$
Since, $i. \ k ≥ 0,$
​​​​​​​$ \therefore k = − 1$ is not possible.
$\therefore\ \text{k}=\frac{1}{10}$
  1. $P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$
$= 0 + k + 2k$
$=3\text{k}=3\times\frac{1}{10}=\frac{3}{10}$
  1. $P(X > 6) = P(X = 7)$
$=7\text{k}^2+\text{k}=7\Big(\frac{1}{10}\Big)^2+\frac{1}{10}=\frac{17}{100}$
  1. $P(0 < X < 3) = P(X = 1) + P(X = 2)$
$= k + 2k = 3k = \frac{3}{10}$

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

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\text{x}^{11}-3\text{x}^9+5\text{x}^7-\text{x}^5+1}{\cos^2\text{x}}\text{ dx}$
→
Evaluate the following integrals:
$\int\frac{(\text{x}^2+1)(\text{x}^2+4)}{(\text{x}^2+3)(\text{x}^2-5)}\ \text{dx}$
→
Differentiate $\tan^{-1}\Big(\frac{1+\text{ax}}{1-\text{ax}}\Big)$ with respect to $\sqrt{1+\text{a}^2\text{x}^2}$
→
Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$
$\text{and G }(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$
Show that
$\big[\text{F}(\alpha)\text{G}(\beta)\big]^{-1}=\text{G}(-\beta)\text{F}(-\alpha).$
→
Prove the following results:
$2\tan^{-1}\Big(\frac{1}{2}\Big)+\tan^{-1}\Big(\frac{1}{7}\Big)=\tan^{-1}\Big(\frac{31}{17}\Big)$
→
Find the area bounded by the lines y = 4x + 5, y = 5 - x and 4y = x + 5.
→
Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
→
ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD, BC and AD. Show that $\overrightarrow{\text{PA}}+\overrightarrow{\text{PB}}+\overrightarrow{\text{PC}}+\overrightarrow{\text{PD}}=4\ \overrightarrow{\text{PQ}}$, where P ia any point.
→
Find the area of a parallelogram ABCD whose side AB and the diagonal AC are given by the vectors $3\hat{\text{i}} + \hat{\text{j}}+4\hat{\text{k}}$ and $4\hat{\text{i}} + 5\hat{\text{k}} $ respectively.
→
Find a vector of magnitude 26 units normal to the plane 12x - 3y + 4z = 1
→