Recent questions and answers in Probability

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Imagine that you do not understand anything in this class. You would have to randomly guess the answer for the above 5 problems. What is the probability that you can get at least two ...
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Two fair 6-face diced are tossed independently. Let X be the random variable of the sum of two numbers on dices and let Y be the absolute difference of two numbers on dices. What is ... X $\geq$ 2Y)?a) 22/36b) 24/36c) 26/36c) 28/36d) 32/36
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You are lost in the National park of Kabrastan. The park population consists of tourists and Kabrastanis. Tourists comprise two-thirds of the population the park and give a correct ... $\left(\dfrac{3}{4}\right)$
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Given the probability mass function $p(X=x)$ ... $.$c=\frac{1}{3}$c=\frac{3}{2}$c=\frac{1}{2}$c=\frac{2}{3}$
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Consider the random variables $(X, Y)$ ... 5,4.5)$.$\frac{1}{3}$\frac{2}{3}$\frac{5}{6}$\frac{1}{2}$
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Consider a discrete random variable $X$ ... >0\right)$ ?$\frac{1}{4}$\frac{3}{8}$\frac{1}{2}$\frac{3}{4}$
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Consider the following discrete random variable problem:A number $X$ is chosen at random from the set $\{1,2,3,4,5\}$. Subsequently, a second number $Y$ is chosen at random from the ... =5 \mid Y=5)=1$\mathbb{P}(X=5, Y=5):=\frac{1}{20}$
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Suppose that $n$ people have their hats returned at random. Let $X_{i}$ be 1 if the $i$ ... n(n-1)}$\frac{2}{n(n-1)}$\frac{1}{n^{2}}$\frac{1}{n}$
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The following is the probability distribution for a random variable, $X$,What is the variance of $X$ ?0.8000.4470.8941.225
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James Bond is playing a game that involves repeatedly rolling a fair standard 6-sided die. What is the expected number of rolls until he gets a 5 ?
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Let $X_{1}$ and $X_{2}$ be two Bernoulli random variables with the probability of success $p$. These variables are independent, where $X_{i}=0$ with probability $1-p$ ...
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Let the joint p.m.f. of $X$ and $Y$ be defined by$f(i, j)=\frac{i+j}{32}$where $i=1,2$ and $j=1,2,3,4$. Find $P(X \leq 3-Y)$.$\frac{5}{32}$\frac{8}{32}$\frac{9}{32}$\frac{10}{32}$
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Let $X$ be a random variable that takes values from 0 to 9 with equal probability $\frac{1}{10}$. Define the random variable $Y=X \bmod 3$.Calculate $P(Y=0)+P(Y=1)$.A. $\frac{3}{10}$B. $\frac{4}{10}$C $\frac{5}{10}$D. $\frac{7}{10}$
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Suppose you are applying to graduate school and want to improve your GRE writing score. Your score on any given day can be either 4 , 5 , or 6 , each with equal probability. You plan to ... $\frac{4}{9}$\frac{5}{9}$\frac{2}{9}$
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Suppose $X$ is the number of dust storms that occur on Mars next year. Assume that $X$ is a discrete uniform random variable that takes one of the 101 values in the range ... frac{1275}{101}$\frac{50}{101}$\frac{100}{101}$\frac{2550}{101}$
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Suppose the discrete random variable $X$ has the probability mass function $f(x)=\frac{11-2 x}{24}, \quad x=1,2,3,4$.Find $E\left[X^{2}\right]$.$\frac{110}{24}$\frac{120}{24}$\frac{130}{24}$\frac{140}{24}$
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Let $X$ and $Y$ be independent discrete random variables with probability mass functions $P(X=$ $k)$ and $P(Y=k)$.What is $P(\min \{X, Y\} \leq x)$ ... x)$P(\min \{X, Y\} \leq x)=P(X>x)+P(Y>x)-P(X>x) P(Y>x)$
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An urn contains $n$ heliotrope and $n$ tangerine balls. A fair die with $n$ sides is rolled. If the $r^{th}$ face is shown, $r$ balls are removed ... a bag. What is the probability that a ball removed at random from the bag is tangerine?
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Consider the following statements:The mean and variance of a Poisson random variable are equal.For a standard normal random variable, the mean is zero and the variance is one. ... $\text{(i)}$ and $\text{(ii)}$ are false
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​​​​​Three fair coins are tossed independently. $T$ is the event that two or more tosses result in heads. $S$ is the event that two or more tosses result in tails.What is the probability of the event $T \cap S$?$0$0.5$0.25$1$
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Let the minimum, maximum, mean and standard deviation values for the attribute income of data scientists be ₹$46000$, ₹ $170000$, ₹ $96000$, and ₹ $21000$, ... $0.476$0.623$2.304$
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​​​​​A fair six-sided die (with faces numbered $1,2,3,4,5,6$ ) is repeatedly thrown independently.What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen?$2$4$6$8$
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Let $X$ be a random variable uniformly distributed in the interval $[1,3]$ and $Y$ be a random variable uniformly distributed in the interval $[2, 4]$. If $X$ and $Y$ are ... $\_\_\_\_\_\_\_\_$ (rounded off to three decimal places).
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Let $X$ be a random variable exponentially distributed with parameter $\lambda>0$. The probability density function of $X$ is given by:\[f_{X}(x)=\left\{\begin{array}{ll}\ ... $\_\_\_\_\_\_\_\_$ (rounded off to one decimal place).
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Consider two events $T$ and $S$. Let $\bar{T}$ denote the complement of the event $T$. The probability associated with different events are given as follows:\[P(\bar{T})=0.6 ... $\_\_\_\_\_\_\_\_$ (rounded off to two decimal places).
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​​​​​​When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers $(i.e., 1, 2, 3, 4, 5, \text{and } 6)$ is$\frac{1}{324}$\frac{5}{324}$\frac{7}{324}$\frac{11}{324}$
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Let $x$ and $y$ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $z=x y$ and let the mean values of $x, y, z$ ... $\bar{z} \leq \bar{x}$
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Consider a permutation sampled uniformly at random from the set of all permutations of $\{1,2,3, \cdots, n\}$ for some $n \geq 4$. Let $X$ be the ... independentEither event $X$ or $Y$ must occurEvent $X$ is more likely than event $Y$
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Let $A$ and $B$ be two events in a probability space with $P(A)=0.3, P(B)=0.5$, and $P(A \cap B)=0.1$. Which of the following statements is/are TRUE? ... $A^c$ and $B^c$ are the complements of the events $A$ and $B$, respectively
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A bag contains $10$ red balls and $15$ blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability (rounded off to $3$ decimal places) that both balls drawn are red is ___________.
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Suppose we have events $A, B$ ... is $\mathrm{P}\left(\mathrm{A}^c \mid \mathrm{B}\right) ?$0.750.550.20.56
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