Probability And Statistics — Part 2 MCQs with Explanations

$$\frac{1}{4}$$ is the correct answer to “A fair dice is rolled twice. The probability that an odd number will follow an even number is”. This question is standard in competitive examination syllabi for Probability And Statistics.

25% is the correct answer to “Two bags A and B have equal number of bails. Bag A has 20% red balls and 80% green bails. Bag B has 30% red bails. 60% green balls and 10% yellow balls. Contents of Bags A and B are mixed thoroughly and a ball is randomly picked from the mixture. What is the chance that the ball picked is red”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{3}$$ is the correct answer to “If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.2666 is the correct answer to “If the standard deviation of the spot speed of vehicles in a highway is 8.8 kmph and the mean speed of the vehicles is 33 kmph, the coefficient of variation in speed is”. This question is standard in competitive examination syllabi for Probability And Statistics.

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

$$\frac{2}{3}$$ is the correct answer to “In a game, two players X and Y are tossing a coin alternately. Whoever gets a 'head' first, wins the game and the game is terminated. Find the chance that player X will win the game if he starts”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{128}}$$ is the correct answer to “A loaded dice has following probability distribution of occurrences Dice Value 1 2 3 4 5 6 Probability $$\frac{1}{4}$$ $$\frac{1}{8}$$ $$\frac{1}{8}$$ $$\frac{1}{8}$$ $$\frac{1}{8}$$ $$\frac{1}{4}$$ If three identical dice as the above are thrown, the probability of occurrence of values 1, 5 and 6 on the three dice is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.5 is the correct answer to “Consider a Gaussian distributed random variable with zero mean and standard deviation σ. The value of its cumulative distribution function at the origin will be”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{9}{{16}}$$ is the correct answer to “Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than $$\frac{1}{2}$$ is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.082 is the correct answer to “An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{\sqrt {12} }}$$ is the correct answer to “The standard deviation of a uniformly distributed random variable between 0 and 1 is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.7 is the correct answer to “X and Y are two random independent events. It is known that P(X) = 0.40 and $${\text{P}}\left( {{\text{X}} \cup {{\text{Y}}^{\text{C}}}} \right) = 0.7.$$ Which one of the following is the value of $${\text{P}}\left( {{\text{X}} \cup {\text{Y}}} \right)\,?$$”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.1497 is the correct answer to “A screening test is carried out to detect a certain disease. It is found that 12% of the positive reports and 15% of the negative reports are incorrect. Assuming that the probability of a person getting a positive report is 0.01, the probability that a person tested gets in incorrect report is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{5}{{12}}$$ is the correct answer to “Suppose afair six-sided die is rolled once. If the value on the die is 1, 2 or 3 the die is rolled a second time. What is the probability that the sum of total values that turn up is at least 6”. This question is standard in competitive examination syllabi for Probability And Statistics.

$${\text{P}}\left( {{\text{r}} + 1} \right) = \frac{{{\text{n}} - {\text{r}}}}{{{\text{r}} + 1}}\frac{{\text{p}}}{{\text{q}}}{\text{P}}\left( {\text{r}} \right)$$ is the correct answer to “If X is a discrete random variable that follows Binomial distribution, then which one of the following response relations is correct”. This question is standard in competitive examination syllabi for Probability And Statistics.

1 and 2 is the correct answer to “The number of parameters in the univariate exponential and Gaussian distributions, respectively, are”. This question is standard in competitive examination syllabi for Probability And Statistics.

R ≥ 0 is the correct answer to “If the difference between the expectation of the square of a random variable (E[x 2 ]) and the square of the expectation of the random variable (E[x]) 2 is denoted by R, then”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{{\text{y}} - {\text{x}}}}$$ is the correct answer to “The probability density function of a continuous random variable distributed uniformly between x and y (for y > x) is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{3}$$ is the correct answer to “If P(x) = $$\frac{1}{4}$$, P(Y) = $$\frac{1}{3}$$ and $${\text{P}}\left( {{\text{X}} \cap {\text{Y}}} \right) = \frac{1}{{12}},$$ the value of $${\text{P}}\left( {\frac{{\text{Y}}}{{\text{X}}}} \right)$$ is”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.222 is the correct answer to “The probability that a communication system will have high fidelity is 0.81. The probability that the system will have both high fidelity and high selectivity is 0.18. The probability that a given system with high fidelity will have high selectivity is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$${\text{P}}\left( {{\text{A}} \cup {\text{B}}} \right) = {\text{P}}\left( {\text{A}} \right) + {\text{P}}\left( {\text{B}} \right)$$ is the correct answer to “Suppose A and B are two independent events with probabilities P(A) ≠ 0 and P(B) ≠ 0. Let $$\overline {\text{A}} $$ and $$\overline {\text{B}} $$ be their complements. Which one of the following statements is FALSE”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{3}{4}$$ is the correct answer to “Probability density function of a random variable X is given below \[{\text{f}}\left( {\text{x}} \right) = \left[ {\begin{array}{*{20}{c}} {0.25}&{{\text{if }}1 \leqslant {\text{x}} \geqslant 5} \\ 0&{{\text{otherwise}}} \end{array}} \right]\,{\text{P}}\left( {{\text{X}} \leqslant 4} \right)\]”. This question is standard in competitive examination syllabi for Probability And Statistics.

$${\left( {\frac{1}{2}} \right)^{\text{N}}}$$ is the correct answer to “A fail coin is tossed N times. The probability that head does not turn up in any of the tosses is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{3}$$ is the correct answer to “There are 25 calculators in a box. Two of them are defective. Suppose 5 calculators are randomly picked for inspection (i.e., each has the same chance of being selected), what is the probability that only one of the defective calculators will be included in the inspection”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{3}{4}$$ is the correct answer to “A single dice is thrown twice. What is the probability that the sum is neither 8 nor 9”. This question is standard in competitive examination syllabi for Probability And Statistics.

Probability (P $$ \cap $$ Q) ≤ Probability (P) is the correct answer to “If P and Q are two random events, then the following is TRUE”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{55}}$$ is the correct answer to “Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled our of the box at random one after another without replacement. The probability that all the three balls are red is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{4}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\int\limits_0^{\text{t}} {{{\text{f}}_1}\left( {\text{x}} \right){{\text{f}}_2}\left( {{\text{t}} - {\text{x}}} \right){\text{dx}}} $$ is the correct answer to “A program consists of two modules executed sequentially. Let f 1 (t) and f 2 (t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.05, 1.87 is the correct answer to “The random variable X takes on the values 1, 2 (or) 3 with probabilities $$\frac{{2 + 5{\text{P}}}}{5},\frac{{1 + 3{\text{P}}}}{5}$$ and $$\frac{{1.5 + 2{\text{P}}}}{5}$$ respectively the values of P and E(X) are respectively.”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{{12}}{{49}}$$ is the correct answer to “There are two containers, with one containing 4 red and 3 green balls and the other containing 3 blue and 4 green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{5}{{16}}$$ is the correct answer to “A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{{\text{e}}}$$ is the correct answer to “Assume that the duration in minutes of a telephone conversion follows the expo-nential distribution $${\text{f}}\left( {\text{x}} \right) = \frac{1}{5}{{\text{e}}^{ - \frac{{\text{x}}}{5}}},\,{\text{x}} \geqslant {\text{0}}{\text{.}}$$ The probability that the conversion will exceed five minutes is”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{1}{2}$$ is the correct answer to “A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is”. This question is standard in competitive examination syllabi for Probability And Statistics.

Mean 33; Median 35; Mode 35 is the correct answer to “Marks obtained by 100 students in an examination are given in the table: Sr. No. Marks obtained No. of students 1 25 20 2 30 20 3 35 40 4 40 20 What would be mean, median and mode of marks obtained by the students”. This question is standard in competitive examination syllabi for Probability And Statistics.

0.5762 is the correct answer to “If X is a normal variable with mean 30 and standard deviation 5, what is probability (26 ≤ X ≤ 34), given A(z = 0.8) = 0.2881”. This question is standard in competitive examination syllabi for Probability And Statistics.

Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different is the correct answer to “If f(x) and g(x) are two probability density functions, \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\text{x}}}{{\text{a}}} + 1}&:&{ - {\text{a}} \leqslant {\text{x}} < 0} \\ { - \frac{{\text{x}}}{{\text{a}}} + 1}&:&{0 \leqslant {\text{x}} \leqslant {\text{a}}} \\ 0&:&{{\text{otherwise}}} \end{array}} \right.;\,\,{\text{g}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} { - \frac{{\text{x}}}{{\text{a}}}}&:&{ - {\text{a}} \leqslant {\text{x}} < 0} \\ {\frac{{\text{x}}}{{\text{a}}}}&:&{0 \leqslant {\text{x}} \leqslant {\text{a}}} \\ 0&:&{{\text{otherwise}}} \end{array}} \right.\] Which one of the following statements is true”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{7}{8}$$ is the correct answer to “If three coins are tossed simultaneously, the probability of getting at least one head is”. This question is standard in competitive examination syllabi for Probability And Statistics.

3 is the correct answer to “Consider a random variable to which a Poisson distribution is best fitted. It happens that P(x = 1) = $$\frac{2}{3}$$P(x = 2) on this distribution plot. The variance of this distribution will be”. This question is standard in competitive examination syllabi for Probability And Statistics.

$$\frac{5}{3}$$ is the correct answer to “A point is randomly selected with uniform probability in the X-Y. plane within the rectangle with corners at (0, 0), (1, 0), (1, 2) and (0, 2). If P is the length of the position vector of the point, the expected value of p 2 is”. This question is standard in competitive examination syllabi for Probability And Statistics.