Probability Set-5

Q1. Two cards are drawn at random without replacement from a standard deck of 52 cards. Given that at least one of the two cards is an ace, what is the probability that both cards are aces?

Q2. A fair die is rolled. If it shows 1 or 2 a coin $C_1$ with $P(H)=\tfrac{1}{4}$ is selected; otherwise a coin $C_2$ with $P(H)=\tfrac{1}{2}$ is selected. The chosen coin is tossed three times and exactly one head is observed. What is the probability that the selected coin was $C_2$?

Q3. The joint pdf of $(X,Y)$ is given by $f_{X,Y}(x,y)=k(x+y)$ for $0<x<y<1$ and $0$ otherwise. Find $P\bigl(X>\tfrac{1}{2}\mid Y>\tfrac{1}{2}\bigr)$.

Q4. Assertion (A): Let $N\sim\text{Poisson}(\lambda)$. Each of the $N$ items is independently classified "red" with probability $p$ and "blue" with probability $1-p$. If $R$ and $B$ denote the counts of red and blue items respectively, then $R$ and $B$ are independent Poisson random variables with means $\lambda p$ and $\lambda(1-p)$.
Reason (R): For $r,s\ge 0$,

so the joint pmf factors as a product of two Poisson pmfs.

Q5. A biased coin with $P(H)=p$ ($0<p<1$) is tossed repeatedly until two consecutive heads appear. Let $T$ be the number of tosses required. The expected value $E[T]$ equals

Q6. A biased coin with probability of heads $p=0.3$ is tossed $5$ times. What is the probability that at least one head appears?

Q7. Three coins are available: a fair coin with $P(H)=\tfrac{1}{2}$, a coin with $P(H)=\tfrac{3}{4}$, and a two-headed coin with $P(H)=1$. One coin is chosen uniformly at random and tossed $4$ times. What is the probability of observing exactly $3$ heads?

Q8. A fair coin is tossed repeatedly until the second head appears. What is the probability that the total number of tosses required is even?

Q9. Four points are chosen independently and uniformly at random on the circumference of a circle. What is the probability that all four points lie on some semicircle of the circle?

Q10. Each element of the set $\{1,2,3,4,5\}$ is included independently with probability $\tfrac{1}{2}$ to form a random subset. What is the probability that the chosen subset contains no two consecutive integers?

Q11. Two fair dice are rolled. Given that at least one die shows $3$, what is the probability that the sum of the two dice is $7$?

Q12. Box I contains $3$ red and $2$ blue balls, and Box II contains $1$ red and $4$ blue balls. A box is chosen with probability $\tfrac{3}{5}$ for Box I and $\tfrac{2}{5}$ for Box II. Two balls are drawn with replacement from the chosen box and both are red. What is the probability that the chosen box was Box I?

Q13. Two cards are drawn at random without replacement from a standard deck of $52$ cards. Given that at least one of the two cards is red, what is the probability that both cards are red?

Q14. A family has two children. Each child is equally likely to be a boy or a girl, and the day of the week each child is born is equally likely (seven days), all independent. Given that at least one child is a boy born on Tuesday, what is the probability that both children are boys?

Q15. Let $X$ and $Y$ be independent random variables uniformly distributed on $[0,1]$. Given that $X+Y>1$, what is the probability that $X>\tfrac{1}{2}$?