Popular mistakes in probability

Question

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, mathematical statistics and random processes?

Some examples of wrong solutions:

Problem 1: Find distribution of $F_{\xi}(\xi)$ for continous $F_{\xi}$.

"Solution": $F_{\xi}(\xi) = P(\xi \le \xi) = 1$.

Problem 2: Is it possible to guess if one of a pair of random numbers is larger with probability ${}> \frac{1}2$?

"Solution". Obviously no (sometimes with some blurry reasoning, mentioning symmetry). (if smb.is interested, see discussion, e.g., in How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?)

Problem 3: Find $\Var(\min(X,Y))$ for independent $X,Y \sim \exp(1)$.

"Solution". If $X \le Y$ then $\min(X,Y) = X$ and $\Var(\min(X,Y)) = DX = 1$, if $X > Y$ then $\min(X,Y) = Y$ and $\Var(\min(X,Y)) = DY = 1$, so in any case we got $\Var(\min(X,Y)) = 1$. A more absurd version is problem 3b: find $D\xi$ for $\xi \sim \Bern(p)$. "Solution": $\xi$ takes values $0$ and $1$, if it's equal to $0$, then $\Var\xi = \Var(0)=0$, in other case $\Var\xi = \Var(1)=0$, hence $\Var(\xi)=0$.

Problem 4: Suppose that systems of sets $\mathcal{A}_1$ and $\mathcal{A}_2$ are independent. Are $\sigma(\mathcal{A}_1)$ and $\sigma(\mathcal{A}_2)$ independent? "Solution". Obviously yes (sometimes with some blurry reasoning).

Problem 5: $2n$ teams were divided to $2$ subgroups with $n$ teams in each group. What is the chance that the $2$ strongest teams will play in the same subgroup?

"Solution": $\frac12$ by symmetry. (See discussion, e.g. Probability problem: what's the chance that two strongest teams will play in the same subgroup.)

Problem 6: "A patient goes to see a doctor. The doctor performs a test with 99 percent reliability—that is, 99 percent of people who are sick test positive and 99 percent of the healthy people test negative. The doctor knows that only 0.01 percent of the people in the country are sick. If the patient test is positive, what are the chances the patient is sick?"

"Solution": 99 percent as follows immediately from the task.

Some references: There are different paradoxes, such as Monty Hall problem, see, e.g. "Paradoxes in Probability Theory and Mathematical Statistics" by G. J. Székely. There are some interesting examples of popular mistakes in "The evolution with age of probabilistic, intuitively based misconceptions" by E. Fischbein and D. Schnarch (and references therein).

Of course, there are a huge number of mistakes that, in principle, can be made, but I mean, firstly, popular ones, and secondly, it is desirable that these were errors not on the most simple topics of combinatorics and not connected with typos and so on. What not-trivial mistakes did you see lots of times?

Addition: uninteresting examples :

1. A coin is tossed twice at random. What is the probability of getting the same face? "Solution:" Three possible outcomes are HH, HT=TH, TT, where H = head, T = tail. So the probability is $\frac13$.

2. $E\xi^2 = (E \xi)^2$ because it's "obvious" because we are speaking about mean, and even reference: there was an equality $E \xi \eta = E\xi E \eta$ in one of the properties of expectation.

3. For independent $\xi$ and $\eta$ we have $\Var(\xi + \eta) = \Var(\xi) + \Var(\eta)$ "hence" $\Var(\xi - \eta) = \Var(\xi) - \Var(\eta)$.

4. Density of $U[0,1]$ is $F' = I_{[0,1]}(x)$ a.e., so density of $\Pois(\lambda)$ is $F'= 0$ a.e.

Answer 1

The most popular mistake made by students is assuming that a sum of uniform random variables is uniform.

A related mistake, done at a more advanced level, is assuming that independence is needed to ensure linearity of the expectation of random variables.