Download App

PART - 2 CH - 14 Probability — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSPART - 2 CH - 14 Probability5 Marks
Question
$A$ and $B$ are throwing two dice one after the other. If before $B$ throwing a 7, $A$ throws 6 then $A$ wins. If $B$ throws a 7 before $A$ throws a 6 then $B$ wins. If $A$ starts throwing then prove that the probability of winning of $A$ is $\frac{30}{61}.$
✓

Answer

Favourable outcomes for obtaining a total of 6 on two dice $=(1,5),(2,4),(3,3),(4,2)$ and $(5,1)$
Probability of getting a sum of $6=\frac{5}{36}$
Probability of not getting a sum of $6=1-\frac{5}{36}=\frac{31}{36}$
In the same way, favourable outcomes for obtaining a sum of 7 are $(1,6),(2,5),(3,4),(4,3),(5,2)$ and $(6,1)$
Probability of getting a sum of $7=\frac{6}{36}=\frac{1}{6}$
Probability of not getting a sum of $6=1-\frac{1}{6}=\frac{5}{6}$
A start then the following possibilities of winning of A are
(i) A gets a 6 in first throw whose probability is $\frac{5}{36}$
(ii) A does not get 6 in first throw, B does not get 7 in second throw and A gets 6 in third throw whose probability will be $\frac{31}{36} \times \frac{5}{6} \times \frac{5}{36}$
(iii) A does not get a 6 in first throw, B does not get 7 in second throw, A does not get 6 in third throw, B does not get a 7 in fourth throw and A gets a 6 in fifth throw whose probability will be
$\left(\frac{31}{36}\right) \times\left(\frac{5}{6}\right) \times\left(\frac{31}{36}\right) \times\left(\frac{5}{6}\right) \times \frac{5}{36}$
$=\left\{\left(\frac{31}{36}\right) \times\left(\frac{5}{6}\right)\right\}^2 \times \frac{5}{36}$
If we assume that A throws dice infinite times to win and since all these events are mutually exclusive, then by addition theorem of probability, probability of winning of $A$ is
$=\frac{5}{36}+\frac{5}{36}\left\{\left(\frac{31}{36}\right) \times\left(\frac{5}{6}\right)\right\}+\frac{5}{36}\left\{\left(\frac{31}{36}\right) \times\left(\frac{5}{6}\right)\right\}^2+\ldots$
The above series is infinite series of geometric progression. Hence sum of infinite terms
$\begin{aligned}=S_{\infty} & =\frac{a}{1-r} \\& =\frac{\frac{5}{36}}{1-\left\{\left(\frac{31}{36}\right)\times\left(\frac{5}{6}\right)\right\}}=\frac{5}{36} \times \frac{36 \times 6}{61} \\& =\frac{30}{61}\end{aligned}$

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 PART - 2 CH - 14 Probability→PART - 2 CH - 14 Probability — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

For any two sets of A and B, prove that:

$\text{A}'\cup\text{B}=\text{U}\Rightarrow\text{A}\subset\text{B.}$

→
Find the distance from the eye at which a coin of 2cm diameter should be held so as to conceal the full moon whose angular diameter is 31'.
→
$\Bigg|\cos\text{x}\cos\Big(\frac{\pi}{3}-\text{x}\Big)\cos\Big(\frac{\pi}{3}+\text{x}\Big)\Bigg|\leq\frac{1}{4}$ for all values of x.
→
Find the equation of the circle which circumscribes the triangle formed by the lines
x + y = 2, 3x - 4y = 6 and x - y = 0.
→
If angle $\theta$ is divided into two parts such that the tangents of one part is $\lambda$ times the tangent of other,and $\phi$ is their difference,the show that $\sin\theta=\frac{\lambda+1}{\lambda-1}\sin\phi$
→
If $\sin(\alpha+\beta)=1$ and $\sin(\alpha-\beta)=\frac{1}{2}$ where $0\leq\alpha,\beta\leq\frac{\pi}{2}$ than find the values of $\tan(\alpha+2\beta)$and $\tan(2\alpha+\beta).$
→
Match the following:
a. If E1 and E2 are the two mutually exclusive events i. $\text{E}_1\cap\text{E}_2=\text{E}_1$
b. If E1 and E2 are the mutually exclusive and exhaustive events ii. $(\text{E}_1-\text{E}_2)\cup(\text{E}_1\cap\text{E}_2)=\text{E}_1$
c. If E1 and E2 have common outcomes, then iii. $\text{E}_1\cap\text{E}_2=\phi,\text{ E}_1\cup\text{E}_2=\text{S}$
d. If E1 and E2 are two events such that $\text{E}_1\subset\text{E}_2$ iv. $\text{E}_1\cap\text{E}_2=\phi$
→
$\text{a}(\cos\text{C}-\cos\text{B})=2(\text{b}-\text{c})\cos^2\frac{\text{A}}{2}.$
→
Prove the following statement by principle of mathematical induction:
$\sqrt{\text{n}}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ ....\ +\frac{1}{\sqrt{\text{n}}},$ for all natural numbers $\text{n}\geq2$
→
Prove that:
$\sin10^\circ\sin50^\circ\sin60^\circ\sin70^\circ=\frac{\sqrt3}{16}$
→