Recent questions tagged conditional-probability

Let $Y$ be a continuous random variable with density,$$f_Y(y)= \begin{cases}2y, & \text { if } 0<y<1 \\\\ 0, & \text { otherwise }\end{cases}$$Given $Y=y$, the conditiona...

Let $X$ be an exponentially distributed random variable with mean $1/\ln 3$. Define $Y=\lfloor X \rfloor$, where $\lfloor X \rfloor$ is the greatest integer less than or ...

A biased coin is flipped $6$ times, and the probability of getting a head on a single flip is $1/3$. Given that the total number of heads is even, what is the probability...

In the anime Jujutsu Kaisen, Kinji Hakari has a Domain Expansion called Idle Death Gamble. His domain works like a gambling machine. It does not directly attack the oppon...

Suppose $X,Y,Z,W$ are independent random variables taking values $0$ or $1$ with equal probability $1/2$. Find $P(X+Y+Z+W=2 \mid X=Y \text{ or } Z=W)$.$1/2$ $1/3$ $1/6$ $...

A delivery worker is assigned one of three types of delivery routes by drawing a card from a box containing $6$ cards numbered $1$ through $6$.If the card is $1$ or $2$, ...

Suppose you keep tossing a fair coin until you get the first head. Let $Y$ be the number of tosses until the first head appears. Find the probability that $Y$ is odd, giv...

Let $X$ and $Y$ be discrete random variables. $X$ takes values $0,1,2,3$ and $Y$ takes values $1,2,3,4$. Their joint probability mass function is given below.$$\begin{arr...

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...

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...

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...

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...

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 ...

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 $A$ 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 $X$ be a continuous random variable with probability density function$$f(x)= \begin{cases}cx^2& , x \in[0,1] \\\\ c(2-x) & , x \in(1,2]\\\\0 & , \text { otherwise }\e...

Suppose $A$ and $B$ are independent events with $0<P(A)<1$ and $0<P(B)<1$. Which of the following statements is/are always correct?$P(A^c\cap B^c)=(1-P(A))(1-P(B))$ $P(A\...