Recent questions tagged goclasses-da-dpp

A number $X$ is chosen from ${1,2,3,4,5,6}$ with $P(X=x)=x/21$.After $X=x$ is chosen, a second number $Y$ is chosen uniformly from ${1,2,\ldots,x}$.Given this setup, whic...

Let $A$, $B$, and $C$ be events such that $P(A)=a$, $P(B\mid A)=b$, $P(B\mid A^c)=c$, $P(C\mid A\cap B)=d$, and $P(C\mid A^c\cap B)=e$. Suppose $C$ can occur only when $B...

Suppose $X$, $Y$, and $Z$ are independent random variables with $E[X]=2$, $E[Y]=-1$, $E[Z]=3$, $\operatorname{Var}(X)=5$, $\operatorname{Var}(Y)=4$, and $\operatorname{Va...

A reliability engineer tests two independent machines. Machine $A$ has $7$ identical safety sensors, and machine $B$ has $11$ identical safety sensors. Each sensor indepe...

In a cyber-security control room, six independent scanners report integer threat scores. Scanner $A$ reports uniformly from $1$ to $5$, scanner $B$ reports uniformly from...

A security system receives $12$ independent login attempts in one minute. Each attempt is routed randomly and equally likely to one of $8$ identical servers. Let $X_i$ be...

Let $X$ and $Y$ be jointly continuous random variables with joint density function : $$f\left(x, y\right)= \begin{cases}cx y(1+y) & \text { if } 0 \leq x \leq 1 \text { a...

For a sample space $S$ containing three events $A$, $B$, and $C$, suppose $P(A)=0.6$, $P(B)=0.5$, and $P(C)=0.4$. Which of the following can NOT be true?Events $A$ and $B...

An analyst is given an anonymized report of the final over of a T20 match. The bowler’s name is hidden. Before seeing the report, the analyst believes the bowler was a de...

A tennis player serves to her opponent's forehand $70\%$ of the time and to the backhand $30\%$ of the time. She wins the point $60\%$ of the time when serving to the for...

A class has $120$ students. There are $64$ students in club $C$, $58$ students in club $T$, and $52$ students in club $M$. Also, $30$ students are in both $C$ and $T$, $2...

In a professional league, $60%$ of players are in Group A and $40%$ are in Group B. In Group A, $2%$ actually use an illegal drug, $8%$ use a legal supplement that can af...

You have three six-sided dice. Die $D_1$ has $1$ red face and $5$ black faces, die $D_2$ has $3$ red faces and $3$ black faces, and die $D_3$ has $5$ red faces and $1$ bl...

There are $12$ cards in a hat. For each rank $1,2,3,4$, there are exactly $3$ cards of that rank. You draw one card at random. If the first card has rank $1$, you draw $1...

Suppose $Z_1,Z_2,Z_3,Z_4$ are i.i.d.(independent & identically distributed) Bernoulli random variables with parameter $p=\frac12$.The random variable, $S=Z_2Z_3+Z_4$ if $...

A login attempt is either genuine or fraudulent. The probability that an attempt is genuine is $0.96$, so the probability that it is fraudulent is $0.04$. If the attempt ...

A binary code $(X,Y,Z)$ is chosen at random from ${0,1}^3$, with all $8$ codes equally likely. Define the events $A={X=Y}$, $B={Y=Z}$, and $C={X=Z}$. Which of the followi...

Let $A$, $B$, and $C$ be events, and assume all conditional probabilities below are defined. Which statement is NOT necessarily true?If $P(A|B)=P(A|B^c)$ and $0<P(B)<1$, ...

Four events $A$, $B$, $C$, and $D$ are mutually independent. Suppose$P(A\cap B\cap C\cap D)=\frac{9}{80}$  $P(A^c\cap B\cap C\cap D)=\frac{27}{160}$  $P(A\cap B^c\cap C\c...

Suppose we want to compute $P(H|E_1,E_2,E_3)$, where no conditional independence assumptions are allowed. Assume all probabilities needed are defined and $0<P(H)<1$. Whic...

Consider three Bernoulli random variables $X$, $Y$, and $Z$, each taking values $0$ or $1$. No independence assumptions are made. Suppose$P(X=1)=\frac{2}{5}$,  $P(Y=1|X=1...

Let $A$, $B$, and $C$ be events. Assume every conditional probability that appears is defined. Which statement is NOT necessarily true?If $0<P(B)<1$ and $P(A|B)>P(A)$, th...

Suppose $X,Y,Z$ are independent binary random variables taking values in ${0,1}$. Let $P(X=0)=p$, $P(Y=0)=q$, and $P(Z=0)=r$. Which expression is equal to $P(X+Y+Z\geq 2\...

Let $A$, $B$, $C$, and $D$ be events. Assume every conditional probability used below is defined. Without making any independence assumptions, which expression(s) must be...

A manufacturer tests one electronic chip using four independent inspection gates. The chip passes Gate A, Gate B, Gate C, and Gate D with probabilities $0.96$, $0.90$, $0...

Let $X$ be a random variable such that $P(X=-2)=\frac{1}{4}$, $P(X=0)=\frac{1}{2}$, and $P(X=2)=\frac{1}{4}$.What is the value of $E[X^2]-{E[X]}^2$ ?

Let $A$ and $B$ be two events with $P(A)>0$ and $P(B)>0$. Suppose $P(A\cap B)$ increases, while $P(B)$ also increases, and $P(A)$ remains unchanged. Which of the followin...

Suppose that $B_1, B_2, \ldots, B_k$ form a partition of the sample space $S$, and $A$ is an event with $0 < P(A) < 1$. Assume also that $P(B_i) 0$ for all $i$. Which of...

Which of the following statements is/are always correct?If $A$ and $B$ are independent and $P(B) 0$, then $P(A^c \mid B) = 1 - P(A)$ If $A$ and $B$ are mutually exclusiv...

Let $S$ be the sample space. Three events $A$, $B$, and $C$ are such that $A$ and $C$ do not overlap, $C \subseteq B$, and $A$ overlaps with $B$. Which of the following s...