Download App

Binomial Distribution — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinomial Distribution4 Marks
Question
A computer installation has 10 terminals. Independently, the probability that anyone terminal will require attention during a week is 0.1. Find the probabilities that (i) 0 (ii) 1 (iii) 2 (iv) 3 or more, terminals will require attention during the next week.
✓

Answer

Let $\mathrm{X}=$ number of terminals which required attention during a week.
$\mathrm{p}=$ probability that any terminal will require attention during a week
$\therefore p=0.1$ and $q=1-p=1-0.1=0.9$
Given: $\mathrm{n}=10$
$\therefore X \sim \mathrm{B}(10,0.1)$
The p.m.f. of $X$ is given by
$\mathrm{P}(\mathrm{X}=\mathrm{x})={ }^n \mathrm{C}_x p^x q^{n-x}$
i.e. $\mathrm{p}(\mathrm{x})={ }^{10} C_x(0.1)^x(0.9)^{10-x}, \mathrm{x}=0,1,2, \ldots, 10$
(i) $\mathrm{P}$ (no terminal will require attention $)=\mathrm{P}(\mathrm{X}=0)$
$ =p(0)={ }^{10} \mathrm{C}_0(0.1)^0(0.9)^{10-0}$
$=1 \times 1 \times(0.9)^{10}=(0.9)^{10} $
Hence, the probability that no terminal requires attention $=(0.9)^{10}$
(ii) $\mathrm{P}(1$ terminal will require attention $)$
$ P(X=1)=p(1)={ }^{10} C_1(0.1)^1(0.9)^{10-1}$
$=10(0.1)(0.9)^9$
$=(1.0)(0.9)^9$
$ =(0.9)^9 $
Hence, the probability that 1 terminal requires attention $=(0.9)^9$.
(iii) $\mathrm{P}(2$ terminals will require attention)
$ P(X=2)=p(2)={ }^{10} \mathrm{C}_2(0.1)^2(0.9)^{10-2}$
$=\frac{10 \times 9}{1 \times 2}(0.1)^2(0.9)^8$
$=45(0.01)(0.9)^8$
$=(0.45) \times(0.9)^8 $
Hence, the probability that 2 terminals require attention $=(0.45)(0.9)^8$.
(iv) $\mathrm{P}$ (3 or more terminals will require attention)
$ =P(X \geqslant 3)$
$=1-P(x<3)$
$=1-[P(X=0)+P(X=1)+P(X=2)]$
$=1-[p(0)+p(1)+p(2)]$
$=1-\left[(0.9)^{10}+(0.9)^9+(0.45)(0.9)^8\right]$
$=1-\left[(0.9)^2+(0.9)^1+0.45\right](0.9)^8$
$=1-[0.81+0.9+0.45](0.9)^8$
$=1-(2.16) \times(0.9)^8 $
Hence, the probability that 3 or more terminals require attention $=1-(2.16) \times(0.9)^8$.

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 "Solve the Following Question.(4 Marks)" from Binomial Distribution→Binomial Distribution — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p
→
Solve the following: A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
→
If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair are also perpendicular.
→
Give the alternative arrangement for the following circuit, so that the new circuit has minimum switches:
Image
→
Let $A(\bar{a})$ and $B(\bar{b})$ are any two points in the space and $R (\overline{ r })$ be a point on the line segment $AB$ dividing it internally in the ratio $m : n$, then prove that $\bar{r}=\frac{m \bar{b}+n \bar{a}}{m+n}$
→
If $\alpha, \beta, \gamma$ are direction angles of line $\ell$, then prove that $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$. Hence deuce that $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=2$.
→
Bismath has half life of 5 days. A sample originally has a mass of $800 \mathrm{mg}$. Find the mass remaining after 30 days.
→
Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0), and (0, 0, c). What is the area of the triangle with these vertices?
→
Find the vector equation of the plane which makes equal non-zero intercepts on the co-ordinates axes and passes through (1, 1, 1).
→
Evaluate: $\int x^2 \sin \ 3 x \ d x$
→