Probability Axioms and Conditioning - Probability - Statistics Previous Year Questions

1

2019 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2019

Let \(\{1,2,3,4\}\) represent the outcomes of a random experiment, and \(P(\{1\}) = P(\{2\}) = P(\{3\}) = P(\{4\}) = 1/4\). Suppose that \(A_1 = \{1,2\}, A_2 = \{2,3\}, A_3 = \{3,4\}\), and \(A_4 = \{1,2,3\}\). Then which of the following statements is true?

A

\(A_1\) and \(A_2\) are not independent.

B

\(A_3\) and \(A_4\) are independent.

C

\(A_1\) and \(A_4\) are not independent.

D

\(A_2\) and \(A_4\) are independent.

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2

2020 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2020

Let E, F and G be mutually independent events with \( P(E) = \frac{1}{2} \), \( P(F) = \frac{1}{4} \) and \( P(G) = \frac{1}{4} \). Let p be the probability that at least two of the events among E, F and G occur. Then \( 12 \times p = \) ___________ (correct up to one decimal place).

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3

2021 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2021

Let $A$ and $B$ be two events such that $P(B) = \frac{3}{4}$ and $P(A \cup B^c) = \frac{1}{2}$. If $A$ and $B$ are independent, then $P(A)$ equals ________ (round off to 2 decimal places).

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4

2021 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2021

A fair die is rolled twice independently. Let $X$ and $Y$ denote the outcomes of the first and second roll, respectively. Then $E(X + Y \mid (X - Y)^2 = 1)$ equals ________

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5

2021 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2021

Let \( \Omega = \{1, 2, 3, ...\} \) represent the collection of all possible outcomes of a random experiment with probabilities \( P(\{n\}) = a_n \) for \( n \in \Omega \). Then which one of the following statements is NOT true?

A

\(\lim_{n \to \infty} a_n = 0 \)

B

\(\sum_{n=1}^\infty \sqrt{a_n} \) converges

C

For any positive integer \(k \), there exist \(k \) disjoint events \(A_1, A_2, ..., A_k \) such that \(P(\cup_{i=1}^k A_i) < 0.001 \)

D

There exists a sequence \(\{A_i\}_{i=1}^\infty \) of strictly increasing events such that \(P(\cup_{i=1}^\infty A_i) < 0.001 \)

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6

2023 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2023

Consider the probability space \( (\Omega, \mathcal{G}, P) \), where \( \Omega = [0, 2] \) and \( \mathcal{G} = \{\phi, \Omega, [0,1), (1,2]\} \). Let \( X \) and \( Y \) be two functions on \( \Omega \) defined as
\[ X(\omega) = \begin{cases} 1 & \text{if } \omega \in [0,1] \\ 2 & \text{if } \omega \in (1,2] \end{cases} \]
and
\[ Y(\omega) = \begin{cases} 2 & \text{if } \omega \in [0,1.5] \\ 3 & \text{if } \omega \in (1.5,2] \end{cases} \]
Then which one of the following statements is true?

A

\(X \) is a random variable with respect to \(\mathcal{G} \), but \(Y \) is not a random variable with respect to \(\mathcal{G} \)

B

\(Y \) is a random variable with respect to \(\mathcal{G} \), but \(X \) is not a random variable with respect to \(\mathcal{G} \)

C

Neither \(X \) nor \(Y \) is a random variable with respect to \(\mathcal{G} \)

D

Both \(X \) and \(Y \) are random variables with respect to \(\mathcal{G} \)

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7

2023 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2023

Two defective bulbs are present in a set of five bulbs. To remove the two defective bulbs, the bulbs are chosen randomly one by one and tested. If \( X \) denotes the minimum number of bulbs that must be tested to find out the two defective bulbs, then \( P(X = 3) \) (rounded off to two decimal places) equals ______________

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8

2023 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2023

Consider the probability space \( (\Omega, \mathcal{G}, P) \), where \( \Omega = \{1, 2, 3, 4\} \), \( \mathcal{G} = \{\phi, \Omega, \{1\}, \{4\}, \{2, 3\}, \{1, 4\}, \{1, 2, 3\}, \{2, 3, 4\}\} \), and \( P(\{1\}) = \frac{1}{4} \). Let \( X \) be the random variable defined on the above probability space as \( X(1) = 1, X(2) = X(3) = 2 \) and \( X(4) = 3 \). If \( P(X \leq 2) = \frac{3}{4} \), then \( P(\{1, 4\}) \) (rounded off to two decimal places) equals __________

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9

2024 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2024

Two fair dice, one having red and another having blue colour, are tossed independently once. Let \( A \) be the event that the die having red colour will show 5 or 6. Let \( B \) be the event that the sum of the outcomes will be 7 and let \( C \) be the event that the sum of the outcomes will be 8. Then which one of the following statements is true?

A

\(A \) and \(B \) are independent as well as \(A \) and \(C \) are independent

B

\(A \) and \(B \) are independent, but \(A \) and \(C \) are not independent

C

\(A \) and \(C \) are independent, but \(A \) and \(B \) are not independent

D

Neither \(A \) and \(B \) are independent, nor \(A \) and \(C \) are independent

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10

2025 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2025

Let \( X \) be a random variable with distribution function \( F \), such that \[ \lim_{h \to 0^+} F(3+h) = \frac{1}{4} \text{ and } F(3) = \frac{3}{4}. \] Then \( 16 \Pr(X = 3) \) equals ______________ (answer in integer).

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11

2026 · Statistics · Probability · Probability Axioms and Conditioning

Statistics (ST) 2026

Let \((X_1, Y_1), (X_2, Y_2), \ldots, (X_n, Y_n), n \geq 2\), be a random sample from a continuous bivariate distribution with joint distribution function \(F_{X,Y}\). Further, \(F_X\) and \(F_Y\) are the marginal distribution functions of \(X\) and \(Y\), respectively. If \(F_{X,Y}(x, y) = F_X(x) F_Y(y), \forall (x, y)\), then, for any two independent pairs \((X_i, Y_i)\) and \((X_j, Y_j)\), \(P[(X_i - X_j)(Y_i - Y_j) > 0]\) equals

A

\(\frac{1}{4}\)

B

\(\frac{1}{2}\)

C

\(\frac{3}{4}\)

D

\(\frac{3}{8}\)

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