Recent questions tagged conditional-probability

Assume $P(A)=0.4$, $P(B)=0.5$, and $P(A \cup B)=0.7$. Also, $P(C)=0.6$, $P(B \cap C) = 0.25$, and and $P(C \mid A \cap B)=\frac{1}{2}$. Find $P(A \mid B \cap C)$.$\frac{2...

In a probability space, it is given that $P(A)=a$, $P(B)=b$, and $P(B \mid A^c)=e$. Also, $P(A \mid B^c)=f$. Determine $P(A \mid B)$ and $P(B \mid A)$ in terms of $a$, $b...

Let $E$ and $F$ be two independent events with $P(E)=\frac{1}{3}$ and $P(F)=\frac{1}{2}$. Let $G$ be an event such that $G \subseteq E'$, $P(G)=\frac{1}{4}$, and $G$ is i...

A college cricket squad has $11$ players consisting of $4$ batsmen, $3$ all-rounders, $3$ bowlers and $1$ wicket keeper. Four players are selected randomly. Given that th...

Let $A$ and $B$ be two events in a sample space such that $P(A' \cap B')=\frac{1}{6}$, $P(A \mid B)=\frac{3}{4}$, and $P(B \mid A')=\frac{2}{5}$. Find $P(A)$.$\frac{11}{1...

Let $A$ and $B$ be two events in a sample space such that $P(A \cup B) = 1$. Given $P(A) = p$ and $P(B) = q$ (where $q 0$), what is the value of $P(A^c \mid B)$ in terms...

Six fair coins are tossed simultaneously and their outcomes are recorded from left to right. Given that at least one pair of consecutive heads occurs, find the probabilit...

An unbalanced die with $6$ faces, numbered $1$ to $6$, is thrown once. The probabilities of obtaining $1$, $3$, and $5$ are equal. The probabilities of obtaining $2$ and ...

A couple has $2$ children. Assume each child is equally likely to be a boy or a girl, independently, and each child is equally likely to be born on any of the $7$ days of...

A fair coin is tossed repeatedly until either the pattern $HTH$ or the pattern $THT$ appears for the first time. What is the probability that $HTH$ appears before $THT$?$...

A student studies for either GATE CS or GATE DA each day. If the student studies GATE CS on a day, then the probability that the student studies GATE CS the next day is $...

Which of the following expressions indicate that $X$ is independent of $Y$ given $Z$?$P(X,Y \mid Z) = P(X \mid Z)P(Y \mid Z)$ $P(X,Y,Z) = P(X,Z)P(Y)$ $P(X \mid Y,Z) = P(X...

Let $X$, $Y$, and $Z$ be discrete random variables with $P(Z=z)>0$. Which of the following statements is equivalent to saying that $X$ and $Y$ are conditionally independe...

In a probabilistic model, suppose $P(A)=a$, $P(B)=b$, and $P(B^c \mid A^c)=e$, where $0<a<1$ and $0<b<1$. Determine $P(A \mid B)$ in terms of $a$, $b$, and $e$.$\frac{b-(...

A number $X$ is chosen from the set ${1,2,3,4,5,6}$ with probability mass function $P(X=x)=cx$ for $x=1,2,3,4,5,6$. Subsequently, given that $X=x$, a second number $Y$ is...

There are $12$ cards in a hat labeled $1,1,1,2,2,2,3,3,3,4,4,4$.You draw one card uniformly at random.If the number drawn is $1$, you draw $1$ more card. If the number dr...

Suppose $Z_1,Z_2,Z_3,Z_4$ are independent and identically distributed Bernoulli random variables with parameter $p=\frac{2}{3}$. Define the random variable $S$ by $S=Z_2+...

In a multiple-choice test, each question has $5$ possible answers. For a given question, a student behaves as follows:With probability $\frac{1}{2}$, the student knows th...

We toss three fair coins simultaneously and independently. Let $A$ be the event that the first and second coins show the same face, $B$ be the event that the second and t...

Consider two Bernoulli random variables $X$ and $Y$, where each can take values $0$ or $1$. Suppose,$$\begin{aligned}& P(X=1)=\frac{2}{5} \\\\& P(Y=1 \mid X=1)=\frac{7}{8...

Which of the following statements is NOT always true?If $A \subseteq B$, then $P(B \mid A)=P(B)-P(A)$. If $P(B)>0$ and $P(A \mid B)=P(A)$, then $A$ and $B$ are independen...

Let $X \in {0,1}$ and $Y \in {0,1}$ be two independent binary random variables. If $P(X=1)=p$ and $P(Y=1)=q$, then $P(X+Y=1 \mid X+Y \ge 1)$ is equal to$\dfrac{p+q-2pq}{p...

In a certain city, the probability that a person will carry an umbrella on a given day is $0.5$. It is known that if the morning is cloudy, then the probability that the ...

Three fair coins are flipped. Given that at least one of the coins shows a head and at least one of the coins shows a tail, what is the probability that exactly two of th...

A medical laboratory buys testing kits from two manufacturers, $A$ and $B$. Manufacturer $A$ supplies $65\%$ of the kits, and manufacturer $B$ supplies $35\%$ of the kits...

One hundred people line up to board an airplane. Each has a boarding pass with assigned seat. However, the first person to board has lost his boarding pass and takes a ra...

There are three doors; behind one is a nice car, and behind each of the other two is a goat eating a bale of straw. You choose a door. Then Monty Hall opens one of the ot...

A multiple choice exam has $4$ choices for each question. The student has studied enough so that the probability they will know the answer to a question is $0.5$, the pro...

Sarah and Bob draw $13$ cards each from a standard deck of $52$ cards. Given that Sarah has exactly two aces, what is the probability that Bob has exactly one ace?$\dfrac...

Consider a clinical test for cancer that can yield either a positive $(+)$ or negative $(-)$ result. Suppose that a patient who truly has cancer has a $1\%$ chance of sli...