Probability Set-3

Q1. Two distinct numbers are selected at random without replacement from the set $\{1,2,\dots,10\}$. What is the probability that their sum is a prime number?

Q2. Let $X\sim\mathrm{Bin}(4,p)$ and $Y\sim\mathrm{Bin}(6,p)$ be independent for some $0<p<1$. Given that $X+Y=3$, find $P\bigl(X=1\mid X+Y=3\bigr)$.

Q3. Three fair dice are rolled. Given that the maximum of the three outcomes is $5$, what is the probability that the sum of the three outcomes is $12$?

Q4. A fair coin is tossed repeatedly until two consecutive heads appear. Let $T$ denote the number of tosses required. Then $\mathbb{E}[T]$ equals:

Q5. Let $X$ and $Y$ be independent random variables uniformly distributed on $[0,1]$. Define events $A=\{|X-Y|\le \tfrac{1}{3}\}$ and $B=\{X\le \tfrac{1}{2}\}$. The conditional probability $P(A\mid B)$ is:

Q6. Two cards are drawn one after another without replacement from a standard deck of 52 cards. What is the probability that both cards are kings?

Q7. One coin is chosen at random from two coins — a fair coin with $P(H)=\tfrac{1}{2}$ and a biased coin with $P(H)=\tfrac{3}{4}$. The chosen coin is tossed twice and results are two heads. What is the probability that the chosen coin was the biased one?

Q8. Two fair dice are rolled. Given that at least one die shows a $4$, what is the probability that the sum of the two dice is $7$?

Q9. A fair coin is tossed repeatedly until two consecutive heads appear. Let $N$ be the number of tosses required. What is the expected value $E(N)$?

Q10. A family has two children. You are told that at least one child is a boy who was born on a Tuesday. What is the probability that both children are boys?

Q11. Bag I contains $3$ red and $2$ white balls and Bag II contains $4$ red and $3$ white balls. A bag is chosen at random and then two balls are drawn from it without replacement. What is the probability that both balls drawn are red?

Q12. Three coins are placed in a box: a fair coin ($P(H)=\tfrac{1}{2}$), a coin with $P(H)=\tfrac{3}{4}$, and a double-headed coin ($P(H)=1$). One coin is chosen at random and tossed three times resulting in three heads. The probability that the chosen coin was the fair coin equals

Q13. Three numbers are chosen at random (without order) from the set $\{1,2,\dots,9\}$. What is the probability that the three chosen numbers form an arithmetic progression?

Q14. A fair coin is tossed repeatedly until two consecutive heads appear. Let $N$ denote the number of tosses required. The expected value $E[N]$ is

Q15. A box contains $10$ coins, of which $3$ are gold and $7$ are silver. Coins are drawn one by one at random without replacement until the first gold coin appears. If $N$ is the draw index of the first gold coin, then $E[N]$ equals