Probability And Statistics | Week 2

Q1. Two cards are drawn successively from a pack without replacing the first. If the first card is spade, then what is the probability that the second card is also a spade?
(A) 13/51
(B) 12/13
(C) 4/17
(D) 3/17

Q2. Four of the seven balls in a box have odd numbers on them. What is the probability that all three balls picked at random, one after the other, without replacement, have odd numbers?
(A) 24/343
(B) 4/35
(C) 2/7
(D) 64/343

Q3. Two urns contain respectively 2 white and 1 black balls, and 1 white and 5 black balls. One ball is transferred from the first to the second urn, and then a ball is drawn from the second urn. What is the probability that the ball drawn is white?
(A) 1/21
(B) 1/3
(C) 4/21
(D) 5/21 – answered by ayush (comments)

Q4. There are 2 boxes : Box-1 contains 3 silver spoons and 3 copper spoons and Box-2 contains 5 copper and 3 silver spoons. Assume that Box-1 is likely to be chosen with a probability of 2/3 and Box-2 is likely to be chosen with a probability of 1/3. A spoon is chosen at random from a box that has been randomly selected. What is the probability that it is a silver spoon?
(A) 11/24
(B) 1/3
(C) 1/8
(D) 5/24

Q5. The chance that a certain disease is diagnosed correctly is 60%. The chance that the patient will die under the treatment, after correct diagnosis is 40%; and the chance of death by wrong diagnosis is 70%. A patient who had the disease died. What is the probability that his/her disease was diagnosed correctly?
(A) 24/25
(B) 6/13
(C) 3/14
(D) 1/6

Q6. Let X be a discrete random variable with the p.m.f. as

Find P(|X|>1)
(A) 1/3
(B) 7/9
(C) 2/9
(D) 8/9

Q7. It is known that 2.5% of mobile phone chargers fail during the warranty period provided they are kept dry. The failure percentage is 5.6, if they are ever wet during the warranty period. If 91% of the chargers are kept dry and 9% are wet during warranty period, what is the probability that a phone charger fails during the warranty period?
(A) 0.3321
(B) 0.4392
(C) 0.0391
(D) 0.0278

Q8. A random variables X has the following probability mass functions
X 0 1 2 3 4 5
P(X=x) 0 2k 3k 2k²  4k 9k²+k

Find P(X < 3).
(A) 0.622
(B) 0.378
(C) 0.455
(D) 0.471

Q9. Suppose the probabilities of n mutually independent events are P₁, P2,…, Pn respectively. Then what is the probability that at least one of the events will occur?
(A) 1- P₁ …. P_n
(B) 1 – (1-P₁) … (1 – P_n)
(C) (1 – P₁) … (1 – P_n)
(D) P₁ …. P_n

Q10. Some elements in a chemical laboratory are highly contaminated. A high amount of contamination in an element is likely to exist with a 0.1 probability. The levels of contamination are evaluated on four randomly chosen elements. How likely is it that at least one of them has a high level of contamination?
(A) 0.2048
(B) 0.4096
(C) 0.6561
(D) 0.7408