Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

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Q. Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2022Subjective· 3mImportance★★★★★

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$X\sim B\left(2,\dfrac16\right)$ (number of sixes in 2 throws); use $E(X)=np$ .

Let $X$ = number of sixes when two dice are thrown. On each die, $P(\text{six})=\dfrac16$ , and the two throws are independent, so $X\sim B\left(2,\dfrac16\right)$ .

$E(X) = np = 2\times\dfrac16 = \dfrac13$

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2023Set ANNUAL2 marksMCQ
$P(X=x) = {}^4C_x\left(\dfrac{5}{9}\right)^x \times \left(\dfrac{4}{9}\right)^{4-x}$
(a) $\dfrac{20}{9}$
(b) $\dfrac{9}{20}$
(c) $\dfrac{12}{9}$
(d) $\dfrac{9}{25}$
›Reveal solutionSolution

This is a Binomial $(4,\frac59)$ p.m.f.; $E(X)=np$ .

The given p.m.f. $P(X=x)={}^4C_x\left(\frac59\right)^x\left(\frac49\right)^{4-x}$ is a Binomial $(n=4,p=\frac59)$ distribution …

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CBSE 2023Set ANNUAL2 marks
$Q. Find the mean of number randomly selected from 1 to 15.$
›Reveal solutionSolution

Mean of a discrete uniform distribution on $\{1,\dots,15\}$ is $\dfrac{1+15}{2}$ .

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CBSE 2022Set ANNUAL2 marksMCQ
$P(x) = \dfrac{c}{x^3}$
(a) $\dfrac{216}{251}$
(b) $\dfrac{294}{251}$
(c) $\dfrac{297}{294}$
(d) $\dfrac{294}{297}$
›Reveal solutionSolution

First find $c$ from $\sum P(x)=1$ , then compute $E(X)=\sum xP(x)$ .

$\sum_{x=1}^3 P(x)=1$ : $\ c\left(1+\dfrac18+\dfrac1{27}\right)=1$

$1+\dfrac18+\dfrac1{27} = \dfrac{216+27+8}{216} = \dfrac{251}{216} \implies c=\dfrac{216}{251}$

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CBSE 2020Set ANNUAL2 marks
$Q. Find expected value of the random variable X whose probability mass function is: X = x: 1, 2, 3; P(X = x): 1/5, 2/5, 2/5.$
›Reveal solutionSolution

$E(X) = \sum x_i P(x_i)$ .

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CBSE 2018Set ANNUAL2 marksMCQ
$P(x) = \dfrac{3-x}{10}$
(a) 1
(b) 2
(c) 0
(d) -1
›Reveal solutionSolution

Compute each probability from the p.m.f., then $E(X)=\sum x\,P(x)$ .

$P(-1)=\frac{3-(-1)}{10}=\frac{4}{10}=0.4,\quad P(0)=\frac{3}{10}=0.3,\quad P(1)=\frac{2}{10}=0.2,\quad P(2)=\frac{1}{10}=0.1$

(Check: $0.4+0.3+0.2+0.1=1$ ✓, a valid p.m.f.)

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CBSE 2017Set ANNUAL2 marksMCQ
$Q. The expected value of the number of heads obtained when three fair coins are tossed simultaneously is$
(a) $1$
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(c) $0$
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›Reveal solutionSolution

$X\sim B(3,\tfrac12)$ ; expected value $E(X)=np$ .

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Q.Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

Concept understanding — Mean (Expectation) of a Random Variable

Q.Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

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