(A) The sum of all probabilities in a probability distribution must be $1$.$\sum P(X=x) = 0.1 + k + 2k + 2k + k = 1$$0.1 + 6k = 1 \implies 6k = 0.9 \i

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(A) The sum of all probabilities in a probability distribution must be $1$.$\sum P(X=x) = 0.1 + k + 2k + 2k + k = 1$$0.1 + 6k = 1 \implies 6k = 0.9 \implies k = 0.15$The probability distribution is:$X$$0$$1$$2$$3$$4$$P(X)$$0.1$$0.15$$0.3$$0.3$$0.15$$1$. Probability of studying at least two hours $(X \geq 2)$:$P(X=2) + P(X=3) + P(X=4) = 0.3 + 0.3 + 0.15 = 0.75$$2$. Probability of studying exactly two hours $(X=2)$:$P(X=2) = 0.3$$3$. Probability of studying at most two hours $(X \leq 2)$:$P(X=0) + P(X=1) + P(X=2) = 0.1 + 0.15 + 0.3 = 0.55$

Class 12 Mathematics Probability Probability distribution

Medium

Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values $x$ has the following form,where $k$ is some unknown constant.
$P(X=x) = \begin{cases} 0.1, & \text{if } x=0 \\ kx, & \text{if } x=1 \text{ or } 2 \\ k(5-x), & \text{if } x=3 \text{ or } 4 \\ 0, & \text{otherwise} \end{cases}$
What is the probability that you study at least two hours? Exactly two hours? At most two hours?

A
$0.75, 0.3, 0.55$
B
$0.55, 0.3, 0.75$
C
$0.3, 0.55, 0.75$
D
$0.75, 0.55, 0.3$

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Probability distribution

If a random variable $X$ has the following probability distribution,then its variance is
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DifficultIIT 1992

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$A$ coin is tossed three times. If $X$ denotes the absolute difference between the number of heads and the number of tails,then $P(X=1) = $

MediumMHT CET 2018

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In a game,a man wins $Rs. 100$ if he gets $5$ or $6$ on a throw of a fair die and loses $Rs. 50$ for getting any other number on the die. If he decides to throw the die either until he gets a five or a six or to a maximum of three throws,then his expected gain/loss (in rupees) is

DifficultJEE MAIN 2019

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If a random variable $X$ has the following probability distribution,then its variance is
$X=x$ $1$ $3$ $5$ $2$
$P(X=x)$ $3 K^2$ $K$ $K^2$ $2 K$