Scoring, pacing and guessing

What Bocconi states, and what is ours

Bocconi's rules give the scoring: $+1$ for a right answer, $0$ for a blank, $-0.25$ for a wrong answer, or $-0.33$ on some questions it does not name. It does not publish how many options each question has, and it gives no advice on guessing or pacing. Everything below the rules is our analysis, using one tool from probability.

Expected value

The expected value of answering is the average number of points the answer would earn if you could repeat the decision many times. With a chance $p$ of being right and a penalty $k$:

$$E = p \times 1 + (1 - p) \times (-k) = p - k(1 - p).$$

A blank always earns $0$, so on average an answer is worth giving when $E > 0$.

A blind guess on a four-option question

Picking one of four options at random gives $p = \frac{1}{4}$. With the $-0.25$ penalty:

$$E = 0.25 - 0.75 \times 0.25 = 0.25 - 0.1875 = 0.0625 = \tfrac{1}{16}.$$

The same in fractions: $\frac{1}{4} - \frac{3}{4} \times \frac{1}{4} = \frac{4}{16} - \frac{3}{16} = \frac{1}{16}$. A blind guess is worth a sixteenth of a point on average: positive, but tiny. It takes about sixteen blind guesses to gain one point on average.

When the penalty or the number of options changes

Because Bocconi names neither the $-0.33$ questions nor the option counts, it is worth seeing how sensitive the answer is:

Options	Penalty	Blind-guess value
4	$-0.25$	$+0.0625$
4	$-0.33$	$0.25 - 0.75 \times 0.33 = +0.0025$
5	$-0.25$	$0.2 - 0.8 \times 0.25 = 0$
5	$-0.33$	$0.2 - 0.8 \times 0.33 = -0.064$
3	$-0.33$	$\frac{1}{3} - \frac{2}{3} \times 0.33 \approx +0.113$

Across these cases a blind guess is worth roughly nothing, sometimes a little more, sometimes a little less. Our working rule is therefore: do not count on blind guessing to gain points. What changes the picture is elimination.

Elimination and the break-even point

Setting $p - k(1 - p) > 0$ and solving gives the break-even chance:

$$p > \frac{k}{1 + k}.$$

For $k = 0.25$ that is $\frac{0.25}{1.25} = 0.2$; for $k = 0.33$ it is $\frac{0.33}{1.33} \approx 0.248$. Ruling out even one option of four with confidence lifts $p$ to $\frac{1}{3}$, above both thresholds:

Four-option question	Value at $-0.25$	Value at $-0.33$
No option ruled out	$+0.0625$	$+0.0025$
One ruled out	$\frac{1}{3} - \frac{2}{3}(0.25) = \frac{1}{6} \approx +0.167$	$\frac{1}{3} - 0.22 \approx +0.113$
Two ruled out	$0.5 - 0.5(0.25) = +0.375$	$0.5 - 0.5(0.33) = +0.335$

So our rule: if you can confidently eliminate at least one option, answer; if you cannot eliminate anything, a blank costs you almost nothing. The word "confidently" matters: if a trap option has drawn you in, your real chance may be lower than you think.

Average is not everything

Expected value describes the long run, and one test is short. Eight blind guesses at $-0.25$ are worth $8 \times 0.0625 = 0.5$ points on average, but the chance that all eight are wrong is $\left(\frac{3}{4}\right)^8 \approx 0.10$, and that outcome costs 2 points. If a few points decide whether you clear the 15-point floor, that spread matters as much as the average.

Pacing a forward-only test

An even pace is 90 seconds per question and $4\frac{1}{2}$ minutes per screen of three. Because questions are mixed by topic and difficulty, a screen may hold a long passage question next to a two-line calculation, so pace by the running total, not by each question. Our suggested checkpoints:

After	Questions done	Clock at most
5 screens	15	22.5 minutes
10 screens	30	45 minutes
15 screens	45	67.5 minutes

There is no second pass, so the usual "skip and return" habit does not exist here. On each screen, give every question a final decision before pressing "Next": an answer, an educated guess after elimination, or a deliberate blank. Our suggestion is a soft cap of about three minutes on any single question; past that, eliminate what you can, decide, and move on. When time runs out the test submits itself, so a question you never reached simply stays blank.