Probability Rules and Conditional Probability - Probability and Statistics - Data Science & Artificial Intelligence Previous Year Questions

Three fair coins are tossed independently. \( T \) is the event that two or more tosses result in heads. \( S \) is the event that two or more tosses result in tails.
What is the probability of the event \( T \cap S \)?

Consider five random variables U, V, W, X, and Y whose joint distribution satisfies:
\(P(U, V, W, X, Y) = P(U)P(V)P(W|U,V)P(X|W)P(Y|W)\)
Which ONE of the following statements is FALSE?

Consider two events \(T\) and \(S\). Let \(\bar{T}\) denote the complement of the event \(T\). The probability associated with different events are given as follows:
\(P(T) = 0.6,\quad P(S|\bar{T}) = 0.3,\quad P(S|T) = 0.6\)
Then, \(P(T|S)\) is ______. (rounded off to two decimal places).

Suppose \(X\) and \(Y\) are random variables. The conditional expectation of \(X\) given \(Y\) is denoted by \(E[X|Y]\). Then \(E[E[X|Y]]\) equals

There are three boxes containing white balls and black balls.
Box-1 contains 2 black and 1 white balls.
Box-2 contains 1 black and 2 white balls.
Box-3 contains 3 black and 3 white balls.
In a random experiment, one of these boxes is selected, where the probability of choosing Box-1 is \(\frac{1}{2}\), Box-2 is \(\frac{1}{6}\), and Box-3 is \(\frac{1}{3}\). A ball is drawn at random from the selected box. Given that the ball drawn is white, the probability that it is drawn from Box-2 is ______________ (Round off to two decimal places)

A random experiment consists of throwing 100 fair dice; each die having six faces numbered 1 to 6. An event \( A \) represents the set of all outcomes where at least one of the dice shows a 1. Then, \( P(A) = \)

Let \( M \) be a randomly chosen non-empty subset of \( S = \{1,2,3, \dots, 2026\} \).
Which of the following is the probability that the product of all the elements of \( M \) is even?

Suppose that a computer program provides a non-negative and integer-valued random solution to the equation \( n_1 + n_2 + n_3 + n_4 = 20 \).
Which of the following is the probability that all of \( n_1, n_2, n_3, n_4 \) in the provided solution are positive?

The number of bijections \( f(\cdot) \) from the set \( S = \{1, 2, 3, 4\} \) to itself such that \( f(f(n)) = n \), for all \( n \in S \), is __________. (Answer in integer)

A clinic specializes in testing for a disease D. The result of the test can be either positive or negative.

A study revealed that if a person suffers from the disease D, the test result in that clinic comes out positive 80% of the time, and negative 20% of the time. If a person is not suffering from the disease D, the test comes out positive 10% of the time and negative 90% of the time. It is also known that among the general population, the disease D occurs in 30% of the individuals.

If a person tests positive for D in that clinic, the probability that he/she actually suffers from the disease D is __________ . (Rounded off to two decimal places)