Probability Set-1

Q1. A bag contains 5 red, 3 blue and 2 green balls. One ball is drawn at random. What is the probability that the ball drawn is either red or green?

Q2. A number is chosen at random from the set $\{1,2,\dots,20\}$. What is the probability that the chosen number is either a prime or a multiple of $3$? (Recall primes $\le 20$ are $2,3,5,7,11,13,17,19$.)

Q3. From a box containing 5 red, 7 blue and 8 green balls, two balls are drawn at random without replacement. What is the probability that both drawn balls are of the same colour?

Q4. Statement A: If $P(A)=0.6$ and $P(B)=0.7$, then $P(A\cap B)\ge 0.3$.
Statement R: $P(A\cap B)=P(A)+P(B)-P(A\cup B)$ and $P(A\cup B)\le 1$, hence $P(A\cap B)\ge P(A)+P(B)-1$.

Q5. Two players alternately draw one ball without replacement from a box containing 2 red and 3 blue balls. The first player draws first and they continue until all balls are drawn. The first player wins if he draws at least one red in his turns (positions 1,3,5). What is the probability that the first player wins?

Q6. A bag contains 3 red, 5 blue and 2 green marbles. Two marbles are drawn at random one after the other without replacement. What is the probability that both marbles drawn are blue?

Q7. A fair coin is tossed three times and a fair six-sided die is rolled once. What is the probability that the number of heads obtained in the three coin tosses equals the number shown on the die?

Q8. Two fair dice are rolled. Given that at least one of the dice shows an even number, what is the probability that the sum of the two dice is a prime number?

Q9. Assertion (A): For two events $A$ and $B$ with $0<P(A),P(B)<1$, if $P(A\mid B)>P(A)$ then $P(B\mid A)>P(B)$.
Reason (R): Because $P(A\mid B)>P(A)$ is equivalent to $P(A\cap B)>P(A)P(B)$, which implies $P(B\mid A)=\dfrac{P(A\cap B)}{P(A)}>P(B)$.

Q10. Three urns $U_1,U_2,U_3$ contain red and blue balls as follows: $U_1$: 2 red, 3 blue; $U_2$: 3 red, 2 blue; $U_3$: 4 red, 1 blue. An urn is chosen at random (each with probability $1/3$) and two balls are drawn from the chosen urn with replacement. Given that the two drawn balls are of different colours, what is the probability that the chosen urn was $U_2$?

Q11. A bag contains 3 red, 2 blue and 5 green balls. Two balls are drawn at random one after the other without replacement. What is the probability that both drawn balls are green?

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

Q13. Two bags are given: Bag I contains $2$ white and $3$ black balls; Bag II contains $4$ white and $1$ black ball. A bag is chosen at random and then two balls are drawn from the chosen bag without replacement. What is the probability that both drawn balls are white?

Q14. A fair coin is tossed. If it shows heads, a fair die is rolled once; if it shows tails, the fair die is rolled twice and the two outcomes are added. What is the probability that the final number obtained (either the single roll or the sum of two rolls) equals $6$?

Q15. An urn contains $4$ white and $6$ black balls. Two players A and B draw alternately one ball each without replacement, starting with A, until a white ball is drawn (the player who draws the first white ball wins). What is the probability that A wins?