Probability Set-2

Q1. Two fair dice are rolled together. What is the probability that the sum of the numbers shown is at least $10$?

Q2. Two fair six-faced dice are rolled. Given that the sum of the two faces is even, what is the probability that the two faces show the same number?

Q3. A bag contains $5$ red and $7$ blue balls. Two balls are drawn at random one after the other without replacement. If it is known that at least one of the drawn balls is red, what is the probability that both drawn balls are red?

Q4. A number is selected at random from the set $\{1,2,\dots,100\}$. What is the probability that the selected number is divisible by $2$ or $3$ but not by $5$?

Q5. Two distinct numbers are chosen at random (without order) from the set $\{1,2,\dots,10\}$. What is the probability that one of the numbers divides the other?

Q6. A bag contains 5 red, 3 blue and 2 green balls. One ball is drawn at random. What is the probability that it is red?

Q7. A number is chosen at random from the set $\{1,2,\dots,10\}$. What is the probability that the chosen number is either a prime number or a multiple of $3$?

Q8. An urn contains 6 white and 4 black balls. Two balls are drawn one after the other without replacement. What is the probability that exactly one of the two drawn balls is white?

Q9. A fair die is rolled repeatedly until a $6$ appears. What is the probability that the number of rolls required is odd?

Q10. A box I contains 2 white and 3 black balls. Box II contains 4 white and 1 black ball. A box is chosen at random and then a ball is drawn from it. If the drawn ball is white, what is the probability that it was drawn from Box II?

Q11. A box contains 10 bulbs, of which 3 are defective and 7 are working. Two bulbs are drawn at random without replacement. What is the probability that both drawn bulbs are working?

Q12. Two fair dice are rolled. Given that at least one of the dice shows a $4$, what is the probability that the sum of the two dice is $7$?

Q13. A box contains 4 white, 3 black and 3 red balls. Two balls are drawn at random without replacement. What is the probability that the two drawn balls are of different colours?

Q14. A fair coin is flipped repeatedly until a head appears; let $X$ be the number of flips needed. A fair six-sided die is rolled once giving a value $D\in\{1,\dots,6\}$. What is the probability that $X\le D$?

Q15. Urn A contains 3 red and 2 white balls; urn B contains 1 red and 4 white balls. One ball is drawn at random from urn A and transferred (without observing its colour) to urn B. Then one ball is drawn at random from urn B and is found to be red. What is the probability that the transferred ball was red?