Probability and Statistics — GATE Previous Year Questions

An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches \(-0.25\) marks. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is:

For each element in a set of size \(2n\), an unbiased coin is tossed. The \(2n\) coin tosses are independent. An element is chosen if the corresponding coin toss is a head. The probability that exactly \(n\) elements are chosen is

Aishwarya studies either computer science or mathematics every day. 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?

An unbalanced die (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is \(90\%\) of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is \(0.75\), which one of the following options is closest to the probability that the face value exceeds 3?

Consider a company that assembles computers. The probability of a faulty assembly of any computer is \(p\). The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of \(q\). What is the probability of a computer being declared faulty?

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

Suppose \(p\) is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and \(p\) has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

Four fair six-sided dice are rolled. The probability of the sum of the results being 22 is \(\dfrac{X}{1296}\). The value of \(X\) is ______.

Suppose \(X_i\) for \(i=1,2,3\) are independent and identically distributed random variables whose probability mass functions are \(\Pr[X_i=0]=\Pr[X_i=1]=1/2\) for \(i=1,2,3\). Define another random variable \(Y=X_1X_2\oplus X_3\), where \(\oplus\) denotes XOR. Then \(\Pr[Y=0\mid X_3=0]=\) ______.

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is \(0.7\), and given that it is of Type 2 is \(0.4\). The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is ______.

\(P\) and \(Q\) are considering to apply for a job. The probability that \(P\) applies for the job is \(\dfrac{1}{4}\), the probability that \(P\) applies for the job given that \(Q\) applies for the job is \(\dfrac{1}{2}\), and the probability that \(Q\) applies for the job given that \(P\) applies for the job is \(\dfrac{1}{3}\). Then the probability that \(P\) does not apply for the job given that \(Q\) does not apply for the job is

Suppose \(Y\) is distributed uniformly in the open interval \((1,6)\). The probability that the polynomial \(3x^{2}+6xY+3Y+6\) has only real roots is (rounded off to 1 decimal place) ______.