Free Printable INTERNAL ASSESSMENT I PROBABILITY AND RANDOM PROCESSES

1.

A coin is tossed up 4 times. The probability that tails turn up in 3 cases is

a)

13

b)

12

c)

14

d)

16

2.

If E denotes the expectation the variance of a random variable X is denoted as

a)

$\left(E\left[X\right]\right)^2$

b)

$2\ E\left(X\right)$

c)

$E\left(X^2\right)$

d)

$E\left(X^2\right)-\left[E\left(X\right)\right]^2$

3.

X is a variate between 0 and 3. The value of $E\left(X^2\right)$ is

a)

9

b)

27

c)

8

d)

4

4.

Three companies A, B and C supply 25%, 35% and 40% of the notebooks to a school. Past experience shows that 5%, 4% and 2% of the notebooks produced by these companies are defective. If a notebook was found to be defective, what is the probability that the notebook was supplied by A?

a)

$\frac{11}{24}$

b)

$\frac{25}{69}$

c)

$\frac{13}{24}$

d)

$\frac{44}{69}$

5.

Two boxes containing candies are placed on a table. The boxes are labelled B₁ and B₂. Box B₁ contains 7 cinnamon candies and 4 ginger candies. Box B₂ contains 3 cinnamon candies and 10 pepper candies. The boxes are arranged so that the probability of selecting box B₁ is ¹⁄₃ and the probability of selecting box B₂ is ²⁄₃. Suresh is blindfolded and asked to select a candy. He will win a colour TV if he selects a cinnamon candy. If he wins a colour TV, what is the probability that the marble was from the first box?

a)

$\frac{13}{7}$

b)

$\frac{7}{33}$

c)

$\frac{7}{13}$

d)

$\frac{6}{33}$

6.

Suppose box A contains 4 red and 5 blue coins and box B contains 6 red and 3 blue coins. A coin is chosen at random from the box A and placed in box B. Finally, a coin is chosen at random from among those now in box B. What is the probability a blue coin was transferred from box A to box B given that the coin chosen from box B is red?

a)

$\frac{14}{29}$

b)

$\frac{1}{2}$

c)

$\frac{7}{10}$

d)

$\frac{15}{29}$

7.

At a certain university, 4% of men are over 6 feet tall and 1% of women are over 6 feet tall. The total student population is divided in the ratio 3:2 in favour of women. If a student is selected at random from among all those over six feet tall, what is the probability that the student is a woman?

a)

$\frac{2}{5}$

b)

$\frac{1}{100}$

c)

$\frac{3}{5}$

d)

$\frac{3}{11}$

8.

If Σ P(x) = k² – 8 then, the value of k is

a)

0

b)

3

c)

12

d)

1

9.

If P(x) = 0.5 and x = 4, then E(x) =

a)

$1$

b)

$0.5$

c)

$2$

d)

$4$

10.

In a discrete probability distribution, the sum of all probabilities is always?

a)

0

b)

Infinite

c)

undefined

d)

1

11.

The expected value of a random variable is its

a)

Mean

b)

Standard Deviation

c)

Variance

d)

None of these

12.

In random experiment, observations of random variable are classified as

a)

Composition

b)

Events

c)

Trials

d)

Functions

13.

The expectation of a random variable X, E(X) can be written as

a)

$\frac{d}{dx}\left[M_x\left(t\right)\right]_{\left(t=0\right)}$

b)

$\frac{d}{dt}\left[M_x\left(t\right)\right]_{\left(t=0\right)}$

c)

$\frac{d^2}{dt^2}\left[M_x\left(t\right)\right]_{\left(t=0\right)}$

d)

$\frac{d^2}{dx^2}\left[M_x\left(t\right)\right]_{\left(t=0\right)}$

14.

If the probability of hitting the target is 0.4, find mean and variance.

a)

0.4, 0.24

b)

0.6, 0.24

c)

0.6, 0.16

d)

0.6, 0.24

15.

If the probability that a bomb dropped from a place will strike the target is 60% and if 10 bombs are dropped, find mean and variance?

a)

0.6, 0.24

b)

0.4, 0.16

c)

4, 1.6

d)

6, 2.4

16.

If P(1) = P(3) in Poisson’s distribution, what is the mean?

a)

$\sqrt{2}$

b)

$\sqrt{6}$

c)

$\sqrt{3}$

d)

$\sqrt{5}$

17.

What is the mean and variance for standard normal distribution?

a)

Mean is 0 and variance is 1

b)

Mean is ∞ and variance is 0

c)

Mean is 0 and variance is ∞

d)

Mean is 1 and variance is 0

18.

Find λ in Poisson’s distribution if the probabilities of getting a head in biased coin toss as 34 and 6 coins are tossed.

a)

3.5

b)

6.6

c)

4.5

d)

5.5

19.

If P(6) = λP(1) in Poisson’s distribution, what is the mean?(Approximate value)

a)

5

b)

6

c)

8

d)

2

20.

Find f(2) in normal distribution if mean is 0 and variance is 1.

a)

0.1468

b)

0.1668

c)

0.1768

d)

0.1568

21.

A number is selected from the first 20 natural numbers. Find the probability that it would be divisible by 3 or 7?

a)

1946

b)

2467

c)

1237

d)

720

22.

If ¹⁶P_r-1 : ¹⁵P_r-1 = 16 : 7 then find r.

a)

10

b)

12

c)

7

d)

8

23.

Find the number of ways of arranging the letters of the words DANGER, so that no vowel occupies odd place.

a)

36

b)

96

c)

144

d)

48

24.

If ⁿP_r = 3024 and ⁿC_r = 126 then find n and r.

a)

9, 4

b)

10, 3

c)

11, 4

d)

12, 4

25.

Mean of a constant ‘a’ is

a)

2

b)

3

c)

a

d)

$\frac{a}{2}$

26.

Variance of a constant ‘a’ is

a)

0

b)

1

c)

a

d)

2

27.

Find the expectation of a random variable X if f(x) = ke^-x for x>0 and 0 otherwise.

a)

0

b)

2

c)

3

d)

1

28.

Find the mean of a random variable X if f(x) = x – ⁵⁄₂ for 0<x<1 and 2x for 1<x<2 and 0 otherwise.

a)

3.5

b)

3.75

c)

2.5

d)

2.75

29.

Find the mean of a continuous random variable X if f(x) = 2e^-x for x>0 and -e^x for x<0.

a)

0

b)

3

c)

2

d)

1

30.

What is moment generating function?

a)

M_x(t) = E(e^tx)

b)

M_x(t) = E(e^-tx)

c)

M_x(t) = E(e^t)

d)

M_x(t) = E(e^2tx)

31.

E(X) = λ is for which distribution?

a)

Bernoulli’s

b)

Poisson’s

c)

Binomial

d)

Normal

32.

E(X) = μ and V(X) = σ² is for which distribution?

a)

Bernoulli’s

b)

Poisson’s

c)

Normal

d)

uniform

33.

The mean of exponential distribution is given as

a)

$\lambda$

b)

$\frac{1}{\lambda^2}$

c)

$\frac{1}{\lambda}$

d)

$\lambda^2$

34.

Consider a random variable with exponential distribution with λ=1. Compute the probability for P (X>3).

a)

e^-3

b)

e^-4

c)

e^-2

d)

e^-1

35.

In badminton practice session, the probability that the player A serves properly is 0.8 and that he player B serves properly is 0.9. If there are only two players, then find the probability that it is serves properly.

a)

0.75

b)

0.85

c)

0.55

d)

0.95

INTERNAL ASSESSMENT I PROBABILITY AND RANDOM PROCESSES

Total questions: 35

Worksheet time: 29mins

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