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QUESTION: 1

Let A and B be any two arbitrary events, then, which one of the following is true?

Solution:

QUESTION: 2

A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is

Solution:

Bag contains 10 white balls and 15 black balls. Required probability

QUESTION: 3

Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is

Solution:

P(atleast one of dice will have 6 facing

= 1 - P(none of dice have 6 facing up)

QUESTION: 4

The probability that top and bottom cards of a randomly shuffled deck are both aces is-

Solution:

The probability that the bottom card of a randomly shuffled deck is ace = 4/52 because there are 4 aces out of total 52 cards.

From the remaining 3 aces out of SVcards the probability that the top card is also an ace = 3/51.

So required probability

QUESTION: 5

E_{1} and E_{2} are events in a probability space satisfying the following constraints:

The value of Pr(E_{1}), the probability of the event E_{1}, is

Solution:

Constraints are

QUESTION: 6

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

Solution:

Let the time taken for first and second modules be represented by x and y and total time = t.

∴ t = x + y is a random variable.

Now the joint density function

which is also called as convolution of f_{1} and f_{2}, abbreviated as f_{1} * f_{2}.

QUESTION: 7

A sample space has two events A and B such that probabilities

What is P(A ∪ B)?

Solution:

We know that:

QUESTION: 8

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

Solution:

where V(x) is the variance of x,

Since variance is and hence never negative,

QUESTION: 9

Consider a random variable Xthat takes values +1 and -1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = -1 and +1 are

Solution:

The p.d.t of the random variable is

The cumulative distribution function F(x) is the probability uptox as given below:

QUESTION: 10

If P and Q are two random events, then the following is TRUE

Solution:

(a) is false since of Pand Q are independent

which need not be zero.

(b) is false since

(c) is false since independence and mutually exclusion are unrelated properties.

(d) is true

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