Probability MCQ

31. If f(x) and g(x) are two probability density functions,

$f(x)=\begin{cases} \frac{x}{a}+1 & \text{ : } -a\leq x< 0 \\ \frac{-x}{a}+1 & \text{ : } 0\leq x\leq a \\ 0 & \text{ : } otherwise \end{cases}\\ g(x)=\begin{cases} \frac{-x}{a} & \text{ : } -a\leq x< 0 \\ \frac{x}{a} & \text{ : } 0\leq x\leq a \\ 0 & \text{ : } otherwise \end{cases}$

Which one of the following statements is true?

1. Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are same
2. Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different
3. Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are same
4. Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are different

32. A two-faced coin has its faces designated as head (H) and tail (T). This coin is tossed three times in succession to record the following outcomes; H, H, H. If the coin is tossed one more time, the probability (up to one decimal place) of obtaining H again, given the previous realizations of H, H and H, would be

1. 1/2
2. 3/2
3. 4/3
4. 5/2

33. An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the second draw is

1. 1/2
2. 4/9
3. 5/9
4. 6/9

34. A six-face fair dice is rolled a large number of times. The mean value of the outcomes is

1. 4.5
2. 3.5
3. 2.5
4. 1.5

35. Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is

1. 1/4
2. 5/2
3. 3/4
4. 4/3

36. Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is μ. The standard deviation for this distribution is given by

1. $\sqrt{\mu}$
2. μ²
3. μ
4. 1/μ

37. The second moment of a Poisson-distributed random variable is 2. The mean of the randor variable is

1. 4
2. 3
3. 2
4. 1

38. The probability that a screw manufactured by a company is defective is 0.1. The company sells screws in packets containing 5 screws and gives a guarantee of replacement if one or more screws in the packet are found to be defective. The probability that a packet would have to be replaced is

39. Type II error in hypothesis testing is

1. acceptance of the null hypothesis when it is false and should be rejected
2. rejection of the null hypothesis when it is true and should be accepted
3. rejection of the null hypothesis when it is false and should be rejected
4. acceptance of the null hypothesis when it is true and should be accepted

40. Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

1. 16/5525
2. 64/2197
3. 3/13
4. 8/16575

41. X and Y are two random independent events. It is known that P(X) =0.40 and P(X ∪ Y^C) = 0.7. Which one of the following is the value of P(X ∪ Y)?

1. 0.7
2. 0.5
3. 0.4
4. 0.3

42. An urn contains 5 red and 7 green balls. A ball is drawn at random and its colour is noted. The ball is placed back into the urn along with another ball of the same colour. The probability of getting a red ball in the next draw is

1. 65/156
2. 67/156
3. 79/156
4. 89/156

43. The probability that a thermistor randomly picked up from a production unit is defective is 0.1. The probability that out of 10 thermistors randomly picked up, 3 are defective is

1. 0.001
2. 0.057
3. 0.107
4. 0.3

44. The probability of obtaining at least two “SIX” in throwing a fair dice 4 times is

1. 125/432
2. 13/144
3. 425/432
4. 19/144

45. Two players, A and B, alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player A starts the game, the probability that A wins the game is

1. 5/11
2. 6/11
3. 1/2
4. 7/13

46. If P(X) = 1/4, P(Y) = 1/3, and P(X ∩ Y) = 1/12, the value of P(Y/X) is

1. 29/50
2. 1/3
3. 1/4
4. 4/25

47. The chance of a student passing an exam is 20%. The chance of a student passing the exam and getting above 90% marks in it is 5%, Given that a student passes the examination, the probability that the student gets above 90% marks is

1. 5/18
2. 1/18
3. 2/9
4. 1/4

48. The number of accidents occurring in a plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month is

1. 0.034
2. 0.029
3. 0.044
4. 0.039

49. A machine produces 0, 1 or 2 defective pieces in a day with associated probability of 1/6, 2/3 and 1/6, respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are

1. 1 and 1/3
2. 1/3 and 1
3. 1 and 4/3
4. 1/3 and 4/3

50. An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

1. 0.067
2. 0.073
3. 0.082
4. 0.091

51. Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is

1. 0.55
2. 0.30
3. 0.60
4. 0.22

52. Consider the following experiment.

Step 1. Flip a fair coin twice.

Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop.

Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.

Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is

(up to two decimal places)

1. 0.66
2. 0.33
3. 0.55
4. 0.76

53. The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is

1. 0.072
2. 0.333
3. 0.064
4. 0.234

54. Two coins R and S are tossed. The 4 joint events H_RH_S, T_RT_S, H_RT_S, T_RH_S have probabilities 0.28, 0.18, 0.30, 0.24, respectively, where H represents head and T represents tail. Which one of the following is TRUE?

1. The Coin tosses are independent
2. R is fair, S is not
3. S is fair, A is not
4. The coin tosses are dependent

55. Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specification are 0.8, 0.7 and 0.5. Each vendor supplies one component. The probability that out of total three components supplied by the vendors, at least one will meet the design specification is

1. 0.90
2. 0.97
3. 0.87
4. 0.67

56. Suppose A and B are two independent events with probabilities P(A) ≠ 0 and P(B) ≠ 0. Let A’ and B’ be their complements. Which one of the following statements is FALSE?

1. P(A ∩ B) = P(A) P(B)
2. P(A/B) = P(A)
3. P(A ∪ B) = P(A) + P(B)
4. P(A’ ∩ B’) = P(A’) P(B’)

57. The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p. Then 100p =

1. 12.90
2. 11.90
3. 13.90
4. 10.90

58. Let S be a sample space and two mutually exclusive events A and B be such that A ∪ B = S. If P(·) denotes the probability of the event, the maximum value of P(A) P(B) is

1. 0.75
2. 0.50
3. 0.25
4. 0.20

59. The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is

60. Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is

1. 14
2. 13
3. 11
4. 10