(B) The number of floors available to leave, excluding the ground floor, is $6$. Each of the $5$ persons can choose any of the $6$ floors independentl

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(B) The number of floors available to leave, excluding the ground floor, is $6$. Each of the $5$ persons can choose any of the $6$ floors independently. Therefore, the total number of ways the $5$ persons can leave is $6^5$. The number of ways the $5$ persons can leave at $5$ different floors is given by the permutation formula $^6P_5$. $^6P_5 = \frac{6!}{(6-5)!} = 6 \times 5 \times 4 \times 3 \times 2 = 720$. The total number of outcomes is $6^5 = 7776$. The probability is $\frac{^6P_5}{6^5} = \frac{720}{7776}$. Simplifying the fraction: $\frac{720}{7776} = \frac{5}{54}$.

Class 12 Mathematics Probability Probability distribution

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$5$ persons entered a lift cabin on the ground floor of a $7$-floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first. The probability of all the $5$ persons leaving the cabin at different floors is

TS EAMCET 2018 All TS EAMCET

A
$\frac{360}{2401}$
B
$\frac{5}{54}$
C
$\frac{5}{18}$
D
$\frac{5!}{7!}$

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Probability

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Probability distribution

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Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0) = \frac{1}{2}$ and $P(X=j) = \frac{1}{3^j}$ for $j = 1, 2, 3, \ldots, \infty$. Then the mean of the distribution and $P(X \text{ is positive and even})$ respectively are:

MediumJEE MAIN 2021

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Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?

Outcome Probability

$\omega_{1}$ $1/8$

$\omega_{2}$ $2/3$

$\omega_{3}$ $1/3$

$\omega_{4}$ $1/3$

$\omega_{5}$ $-1/4$

$\omega_{6}$ $-1/3$

Outcome	Probability
$\omega_{1}$	$1/8$
$\omega_{2}$	$2/3$
$\omega_{3}$	$1/3$
$\omega_{4}$	$1/3$
$\omega_{5}$	$-1/4$
$\omega_{6}$	$-1/3$

Easy

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The p.d.f. of a continuous random variable $X$ is given by $f(x) = \frac{1}{2}$ if $0 < x < 2$ and $f(x) = 0$ otherwise. If $a = P(X < \frac{1}{2})$ and $b = P(X > \frac{3}{2})$,then the relation between $a$ and $b$ is:

EasyMHT CET 2020

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$A$ biased coin with probability $p, 0 < p < 1,$ of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $\frac{2}{5},$ then $p = $

Difficult

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If the probability distribution of a random variable $X$ is as follows,then $P(X \leq 2) = $
$x_i$ $0$ $1$ $2$ $3$ $4$
$P(X = x_i)$ $3K$ $5K$ $3k^2$ $4k^2 + k$ $3k^2$

$x_i$	$0$	$1$	$2$	$3$	$4$
$P(X = x_i)$	$3K$	$5K$	$3k^2$	$4k^2 + k$	$3k^2$

MediumAP EAMCET 2025

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$5$ persons entered a lift cabin on the ground floor of a $7$-floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first. The probability of all the $5$ persons leaving the cabin at different floors is

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Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0) = \frac{1}{2}$ and $P(X=j) = \frac{1}{3^j}$ for $j = 1, 2, 3, \ldots, \infty$. Then the mean of the distribution and $P(X \text{ is positive and even})$ respectively are:

Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid? Outcome Probability $\omega_{1}$ $1/8$ $\omega_{2}$ $2/3$ $\omega_{3}$ $1/3$ $\omega_{4}$ $1/3$ $\omega_{5}$ $-1/4$ $\omega_{6}$ $-1/3$

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Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?

Outcome Probability

$\omega_{1}$ $1/8$

$\omega_{2}$ $2/3$

$\omega_{3}$ $1/3$

$\omega_{4}$ $1/3$

$\omega_{5}$ $-1/4$

$\omega_{6}$ $-1/3$

If the probability distribution of a random variable $X$ is as follows,then $P(X \leq 2) = $
$x_i$ $0$ $1$ $2$ $3$ $4$
$P(X = x_i)$ $3K$ $5K$ $3k^2$ $4k^2 + k$ $3k^2$