Probability Set-2

Q1. A fair six-faced die is rolled twice. Given that the sum of the two outcomes is $7$, what is the probability that the first roll is $4$?

Q2. Let $X$ and $Y$ be independent random variables uniformly distributed on $(0,1)$. Find the conditional probability $P\!\big(X+Y<\tfrac{1}{2}\mid \max\{X,Y\}>\tfrac{1}{3}\big)$.

Q3. A fair six-sided die is rolled repeatedly until a $6$ appears for the first time. Let $N$ be the number of rolls required. What is $P(N\ \text{is even})$?

Q4. Assertion (A): Let $X\sim\text{Pois}(\lambda_1)$ and $Y\sim\text{Pois}(\lambda_2)$ be independent. Then the conditional distribution of $X$ given $X+Y=n$ is $\text{Bin}\!\big(n,\tfrac{\lambda_1}{\lambda_1+\lambda_2}\big)$.
Reason (R): The Poisson distribution arises as the limit of $\text{Binomial}(m,p)$ when $m\to\infty$, $p\to0$ with $mp\to\lambda$.

Q5. Let $N\sim\text{Pois}(\lambda)$ and, given $N$, let $X\mid N\sim\text{Bin}(N,p)$ (each of the $N$ trials succeeds independently with probability $p$). Find $\operatorname{Var}(X)$.

Q6. A biased coin with probability of head $0.4$ is tossed independently $5$ times. What is the probability of getting at least one head?

Q7. The number of emails arriving in an hour follows a Poisson distribution with parameter $\lambda$. Given that $P(0)=2P(1)$, find $P(\text{at least }2\ \text{emails})$.

Q8. A box contains three coins: a fair coin ($P(H)=\tfrac{1}{2}$), a two-headed coin (always heads), and a biased coin with $P(H)=\tfrac{3}{4}$. One coin is chosen uniformly at random and tossed twice; both tosses result in heads. What is the probability that the chosen coin is the two-headed coin?

Q9. Let $(X,Y)$ have joint pdf $f(x,y)=k(x+y)$ for $0<x<1,\;0<y<1$ and $f(x,y)=0$ otherwise. Given that $X+Y>1$, find $P\bigl(X>\tfrac{1}{2}\mid X+Y>1\bigr)$.

Q10. A fair six-sided die is rolled repeatedly until the first '6' appears. Let $N$ be the trial number on which the first 6 occurs. What is $P(N\ \text{is even})$?