Probability Set-1

Q1. A box contains $5$ balls, $3$ red and $2$ blue. Three balls are drawn at random without replacement. What is the probability that exactly two of the drawn balls are red?

Q2. Bag I contains $5$ red and $3$ blue balls, and Bag II contains $3$ red and $7$ blue balls. A bag is chosen with probabilities $P(\text{Bag I})=\dfrac{2}{5}$ and $P(\text{Bag II})=\dfrac{3}{5}$, then two balls are drawn from the chosen bag without replacement. What is the probability that both drawn balls are red?

Q3. A $4$-digit number is formed by choosing digits from $0$ to $9$ without repetition and the first digit cannot be $0$. If a number is chosen uniformly at random from all such $4$-digit numbers with distinct digits, what is the probability that it is divisible by $5$?

Q4. Real numbers $a$ and $b$ are chosen independently and uniformly from $[0,1]$. For the quadratic equation $x^{2}+ax+b=0$ to have real roots we require $a^{2}-4b\ge 0$. What is the probability that the equation has real roots?

Q5. A stick of unit length is broken at two points chosen independently and uniformly along its length, producing three pieces. What is the probability that the three pieces can form the sides of a triangle?

Q6. (actually Que51) A biased coin has probability of head $0.6$. It is tossed $5$ times. The probability of getting exactly $3$ heads is

Q7. (actually Que52) Box I contains $4$ white and $6$ black balls, Box II contains $7$ white and $3$ black balls. A box is chosen with probabilities $0.6$ (Box I) and $0.4$ (Box II). From the chosen box two balls are drawn with replacement and both are white. The probability that the chosen box was Box I equals

Q8. (actually Que53) An urn contains $10$ bulbs, of which $3$ are defective and $7$ are good. Three bulbs are chosen at random without replacement. Given that at least one of the selected bulbs is defective, the probability that exactly two selected bulbs are defective is

Q9. (actually Que54) Three independent Bernoulli trials have success probability $p$ each. Let $X$ be the number of successes in the first two trials and $Y$ be the number of successes in the last two trials. The correlation coefficient $\rho(X,Y)$ is

Q10. (actually Que55) Let $X\sim\mathrm{Bin}(5,p)$ and $Y\sim\mathrm{Bin}(3,p)$ be independent. Given that $X+Y=6$, the conditional probability $P(X=4\mid X+Y=6)$ equals