Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Would you have fallen for these?

Each is a real option from a previous-year paper. Read it cold and decide before opening it.

• Option C

  4/25

  Why this is wrong, and why it looks right

  $4/25 = (4/10)^{2}$ treats the second draw as independent of the first, as if the ball were replaced — so 4 reds out of 10 are available both times. Without replacement, one red gone leaves only 3 red among 9.

  From this question

  A bag has 4 red and 6 blue balls. Two are drawn without replacement. The probability both are red is

  Part A · free

• Option A

  $X_{(1)}$ and $X_{(4)}$ are independent.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · june 2023

• Option D

  $2\chi^{2}_{4}$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · june 2023

• Option B

  $Cov(\varepsilon$̂$_{i}, \varepsilon$̂$_{k}) = Cov(\varepsilon_{i}, \varepsilon_{k})$, for all i ≠ k = 1, 2, …, n

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option C

  Var(Ŷ$_{i}) = Var(Y_{i}), 1 \le i \le n$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option A

  90

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option D

  $U_{(1)}$ and $U_{(5)}$ are independent.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part C

• Option A

  $X_{n}$ converges to X in probability as $n \to \infty$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part C · december 2024

• Option B

  $E(Z^{2}) = 2$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part C · december 2024

• Option B

  Cov(Ŷ$_{i}$, Ŷ$_{k}) = Cov(Y_{i}, Y_{k}), 1 \le i < k \le n$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option C

  $Var(\varepsilon$̂$_{i}) = Var(\varepsilon_{i}), 1 \le i \le n$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option D

  Pairwise independence of three events implies their mutual independence.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part C

• Option A

  $P(A_{1}$ | $A_{2})P(A_{2}$ | $A_{3}) \le P(A_{1}$ | $A_{3})$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · june 2024

• Option B

  $P(A_{1}$ | $A_{2})P(A_{3}$ | $A_{2}) \ge P(A_{1} \cap A_{3}$ | $A_{2})$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · june 2024

• Option D

  $P(A_{1}$ | $A_{2}) + P(A_{2}$ | $A_{3}) \le P(A_{1}$ | $A_{3})$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · june 2024

• Option D

  $Cov(\ln(X_{(1)}) - \ln(X_{(1)} + X_{(n)}), X_{(n)}) < 0$

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part C · june 2024

• Option B

  B is a closed subset of $\mathbb{R}^{2}$ but A is not a closed subset of $\mathbb{R}^{2}$.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option C

  Both A and B are closed subsets of $\mathbb{R}^{2}$.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option D

  Neither A nor B is a closed subset of $\mathbb{R}^{2}$.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option D

  For random variables in general (not jointly normal), zero correlation implies independence.

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part C

• Option D

  2/25

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part A · december 2025

• Option B

  0.5

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option C

  −1

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option D

  1

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · december 2024

• Option A

  1/2

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B · june 2025

• Option C

  0.3

  Why this is wrong, and why it looks right

  The option is above and the answer is on the question page, both free. This is the part that is not: the reasoning that makes this particular wrong option feel like the right one.

  Part B

The counterexamples that settle these

One example is enough to kill a false statement. These are the standard ones for the claims above.

• Convergence in probability ⇒ almost sure convergence
• Pairwise independent events are mutually independent
• The maximum of independent exponentials is exponential
• If X and Y are each normal and uncorrelated then they are independent
• Uncorrelated implies independent
• Convergence in probability implies almost sure convergence

Other mistakes that cost marks