Probability And Statistics — Part 1 MCQs with Explanations

0.4 is the correct answer to “Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that she studies mathematics the next day is 0.6. If she studies mathematics on a day, then the, probability that she studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{4}$$ is the correct answer to “A coin is tossed 4 times. What is the probability of getting heads exactly 3 times”. This question is standard in competitive examination syllabi for Probability And Statistics.

1 and $$\frac{1}{3}$$ is the correct answer to “A machine produces 0, 1 or 2 defective pieces in a day with associated probability of $$\frac{1}{6},\,\frac{2}{3}$$ and $$\frac{1}{6}$$ respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are”. This question is standard in competitive examination syllabi for Probability And Statistics.

E(X2Y2) = (E(X))2 (E(Y))2 is the correct answer to “Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{{19}}{{25}}$$ is the correct answer to “The box 1 contains chips numbered 3, 6, 9, 12 and 15. The box 2 contains chips numbered 6, 11, 16, 21 and 26. Two chips, one from each box, are drawn at random. The numbers written on these chips are multiplied. The probability for the product to be an even number is”. This question is standard in competitive examination syllabi for Probability And Statistics.

1 - 2α is the correct answer to “For random variable x($$ - \infty $$ < x < $$\infty $$) following normal distribution, the mean is μ = 100. If probability is P = α for x ≥ 110. Then probability of x lying between 90 and 110 i.e., P(90 ≤ x ≤ 110) is equal to”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{2}{9}$$ is the correct answer to “A box contains 5 black and 5 red balls. Two balls are randomly picked one after another from the box, without replacement. The probability for both balls being red is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.6 is the correct answer to “In the following table, x is a discrete random variable and p(x) is the probability density. The standard deviation of x is x 1 2 3 p(x) 0.3 0.6 0.1”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.4 is the correct answer to “Three companies X, Y and Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below: Company % of computer Probability of being supplied defective X 60% 0.01 Y 30% 0.02 Z 10% 0.03 Given that a computer is defective, the probability that it was supplied by Y is”. This question is standard in competitive examination syllabi for Probability And Statistics.

exponentially distributed with mean $$\frac{1}{6}$$ is the correct answer to “Let X 1 and X 2 be two independent exponentially distributed random variables with means 0.5 and 0.25 respectively. Then Y = min (X 1 , X 2 ) is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{1260}}$$ is the correct answer to “A box contains 2 washers, 3 nuts and 4 bolts. Items are drawn from the box at random one at a time without replacement. The probability of drawing 2 washers first followed by 3 nuts and subsequently the 4 bolts is”. This question is standard in competitive examination syllabi for Probability And Statistics.

9375 is the correct answer to “An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability, The sum total of the expected marks obtained by all these students is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{3}$$ is the correct answer to “The graph of a function f(x) is shown in the figure. For f(x) to be valid probability density function, the value of h is”. This question is standard in competitive examination syllabi for Probability And Statistics.

17 is the correct answer to “A sample of 15 data is as follows: 17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. The mode of the data is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.79 is the correct answer to “What is the probability that at most 5 defective fuses will be found in a box of 200 fuses, if 2% of such fuses are defective”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{169}}$$ is the correct answer to “Two cards are drawn at random in succession with replacement from a deck of 52 well shuffled card. Probability of getting both aces is”. This question is standard in competitive examination syllabi for Probability And Statistics.

< 50% is the correct answer to “The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is”. This question is standard in competitive examination syllabi for Probability And Statistics.

μ - 1 is the correct answer to “The function p(x) is given by $${\text{p}}\left( {\text{x}} \right) = \frac{{\text{A}}}{{{{\text{x}}^\mu }}}$$ where A and μ are constants with μ > 1 and 1 ≤ x < $$\infty $$ and p(x) = 0 for $$ - \infty $$ < x < 1. For p(x) to be a probability density function, the value of A should be equal to”. This question is standard in competitive examination syllabi for Probability And Statistics.

$${\left( {\frac{1}{2}} \right)^{10}}$$ is the correct answer to “A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.034 is the correct answer to “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”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.040 is the correct answer to “A hydraulic structure has four gates which operate independently. The probability of failure of each gate is 0.2. Given that gate 1 has failed, the probability that both gates 2 and 3 will fail is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{5}{{12}}$$ is the correct answer to “A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is”. This question is standard in competitive examination syllabi for Probability And Statistics.

σy = aσx + b is the correct answer to “Consider the finite sequence of random values X = [x 1 , x 2 , ... , x n ]. Let μ x be the mean and σ x be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as y i = a ∗ x i + b, where a and b are positive constant. Let μ y be the mean and σ y be the standard deviation of this sequence. Which one of the following statements INCORRECT”. This question is standard in competitive examination syllabi for Probability And Statistics.

Pr(r = 8 | r/4 is an integer) = (5/9) is the correct answer to “Two fair dice are rolled and the sum r of the numbers turned up is considered”. This question is standard in competitive examination syllabi for Probability And Statistics.

The coin tosses are dependent is the correct answer to “Two coins R and S are tossed. The 4 joint events H R H S , T R T S , H R T S , T R H 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”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.5 is the correct answer to “If {x} is a continuous, real valued random variable defined over the interval (-$$\infty $$, +$$\infty $$) and its occurrence is defined by the density function given as: $${\text{f}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}$$ where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$”. This question is standard in competitive examination syllabi for Probability And Statistics.

acceptance of the null hypothesis when it is false and should be rejected is the correct answer to “Type II error in hypothesis testing is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\left\{ {\frac{1}{2},\,2} \right\}$$ is the correct answer to “A random variable X has a probability density function, \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {{\text{k}}{{\text{x}}^{\text{n}}}{{\text{e}}^{ - {\text{x}}}};}&{{\text{x}} \geqslant {\text{0}}} \\ {0;}&{{\text{otherwise}}} \end{array}} \right.\] (n is an integer) with mean 3. The values of {k, n} are”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.33 is the correct answer to “A six-faced fair dice is rolled five times. The probability (in%) of obtaining "ONE" at least four times is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.5$$\alpha $$ is the correct answer to “A probability density function is of the form $${\text{p}}\left( {\text{x}} \right) = {\text{K}}{{\text{e}}^{ - \alpha \left| x \right|}},\,{\text{x}} \in \left( { - \infty ,\,\infty } \right).$$ The value of K is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{{19}}{{495}}$$ is the correct answer to “A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective”. This question is standard in competitive examination syllabi for Probability And Statistics.

9.0 is the correct answer to “A class of first year B. Tech. students is composed of four batches A, B, C and D, each consisting of 30 students. It is found that the sessional marks of students in Engineering Drawing in batch Chave a mean of 6.6 and standard deviation of 2.3. The mean and standard deviation of the marks for the entire class are 5.5 and 4.2, respectively. It is decided by the course instructor to normalize the marks of the students of all batches to have the same mean and standard deviation as that of the entire class. Due to this, the marks of a student in batch C are changed from 8.5 to”. This question is standard in competitive examination syllabi for Probability And Statistics.

greater than zero and less than 0.5 is the correct answer to “Let X be a normal random variable with mean 1 and variance 4. The probability P{X < 0} is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{{16}}{{5525}}$$ is the correct answer to “Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{6}$$ is the correct answer to “A committee of 4 is to be formed from among 4 girls and 5 boys. What is the probability that the committee will have number of boys less than number of girls”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.062 is the correct answer to “It is estimated that average number of events during an year is three. What is the probability of occurrence of not more than two events over a two year duration? Assume that the number of events follow a Poisson's distribution.”. This question is standard in competitive examination syllabi for Probability And Statistics.

None of these is the correct answer to “Suppose we uniformly and randomly select a permutation from the 20! permutations of 1, 2, 3 ...., 20. What is the probability that 2 appears at an earlier position that any other even number in the selected permutation”. This question is standard in competitive examination syllabi for Probability And Statistics.

50.0% is the correct answer to “The standard normal cumulative probability function (probability from $$ - \infty $$ to x n ) can be approximated as $${\text{F}}\left( {{{\text{x}}_{\text{n}}}} \right) = \frac{1}{{1 + {\text{exp}}\left( { - 1.7255{{\text{x}}_{\text{n}}}{{\left| {{{\text{x}}_{\text{n}}}} \right|}^{0.12}}} \right)}}$$ where x n = standard normal deviate. If mean and standard deviation of annual precipitation are 102 cm and 27 cm respectively, the probability that the annual precipitation will be between 90 cm and 102cm is”. This question is standard in competitive examination syllabi for Probability And Statistics.

Mean is greater than median and mode is the correct answer to “A probability distribution with right skew is shown in the figure. The correct statement for the probability distribution is”. This question is standard in competitive examination syllabi for Probability And Statistics.

3 is the correct answer to “Let X be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If P(X ≤ -1) = P(Y ≥ 2) the standard deviation of Y is”. This question is standard in competitive examination syllabi for Probability And Statistics.