The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 7 is:

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:

For n>2, let a ∈ {0,1}ⁿ be a non-zero vector. Suppose that x is chosen uniformly at random from {0,1}ⁿ. Then, the probability that  is an odd number is _____.

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

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

The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that it will rain either today or tomorrow is 0.7. What is the probability that it will rain today and tomorrow?

Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial is __________.

A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is

Consider two events E₁ and E₂ such that probability of E₁, Pr[E₁]=1/2, probability of E₂, Pr[E₂]=1/3, and probability of E₁ and E₂, Pr[E₁ and E₂]=1/5. Which of the following statement is/are True?

E₁ and E₂ are events in a probability space satisfying the following constraints:

• Pr(E₁) = Pr(E₂)
• Pr(E₁ U E₂) = 1
• E₁ and E₂ are independent

The value of Pr(E₁), the probability of the event E₁ is

Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day?

Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is

Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A|B) and P(B|A) respectively are

A program consists of two modules executed sequentially. Let f₁(t) and f₂(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:

If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

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:

Two n bit binary strings, S₁ and S₂, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

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 p² is

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

The probability of an event B is P₁. The probability that events A and B occur together is P₂ while the probability that A and occur together is P₃. The probability of the event A in terms of P₁, P₂ and P₃ is __________.

Let A, B and C be independent events which occur with probabilities 0.8, 0.5 and 0.3 respectively. The probability of occurrence of at least one of the event is __________

Let X and Y be two exponentially distributed and independent random variables with mean α and β, respectively. If Z = min(X,Y), then the mean of Z is given by

In how many ways can we distribute 5 distinct balls, B₁, B₂, …, B₅ in 5 distinct cells, C₁, C₂, …, C₅ such that Ball B, is not in cell C_i, ∀i = 1, 2, …, 5 and each cell contains exactly one ball?

A bag contains 10 blue marbles, 20 green marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag. This process is repeated 3 times. The probability that no two of the marbles drawn have the same colour is

An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the trials are independent, the expected number of tosses is

In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data indicates that if the temperature at noon is less than or equal to 25°C, the probability that it will rain in the afternoon is 0.4. The temperature at noon is equally likely to be above 25°C, or at/below 25°C. What is the probability that it will rain in the afternoon on a day when the temperature at noon is above 25°C?

When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that successive tosses are independent, the expected value of N is

Suppose there are two coins. The first coin gives heads with probability 5/8 when tossed, while the second coin gives heads with probability 1/4. One of the two coins is picked up at random with equal probability and tossed. What is the probability of obtaining heads ?

Consider a hash function that distributes keys uniformly. The hash table size is 20. After hashing of how many keys will the probability that any new key hashed collides with an existing one exceed 0.5.

In a multi-user operating system on an average, 20 requests are made to use a particular resource per hour. The arrival of requests follows a Poisson distribution. The probability that either one, three or five requests are made in 45 minutes is given by :

A sample space has two events A and B such that probabilities P(A ∩ B) = 1/2, P(A') = 1/3, P(B') = 1/3. What is P(A ∪ B)?

What is the probability that in a randomly chosen group of r people at least three people have the same birthday?

Which of the following statements are FALSE?

Let f(x) be the continuous probability density function of a random variable X. The probability that a < X ≤ b, is:

Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows:

(i) Select a box
(ii) Choose a ball from the selected box such that each ball in
     the box is equally likely to be chosen. The probabilities of
     selecting boxes P and Q are (1/3) and (2/3), respectively.

Given that a ball selected in the above process is a red ball, the probability that it came from the box P is

A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that two such randomly generated strings are not identical is:

P and Q are considering to apply for job. The probability that p applies for job is 1/4. The probability that P applies for job given that Q applies for the job 1/2 and The probability that Q applies for job given that P applies for the job 1/3.The probability that P does not apply for job given that Q does not apply for the job