Over half a million students in Italy today faced the Second test of the Maturity 2025. Cicero’s brace, present in classic high schools with the De Amicitia and in scientific high schools with quote from De divination which appears in one of the 8 questions of the Mathematics test. The topic? The so -called “Shot of Venus”that is, the situation in which 4 4 -faced dice are launched obtaining 4 different results. In the era of ancient Rome, the most lucky dice configuration was considered. No small fact, considering that at the time the astragal (i.e. the dice) were used for the divination as well as for gambling. But how probable is it? This is exactly what the question asks: “Assuming that the faces of each dice are can be effectively, determine the probability of obtaining Venus’s blow in the launch of 4 dice and the probability of obtaining 4 numbers all equal”. Here is the solution to this question on probability!
Let’s start with Mathematical definition of probabilitythat is, the relationship between the number of favorable cases (in this case all the configurations in which the numbers are all different) and total cases (in this case all the possible configurations of the 4 dice after the launch).
For the number of favorable cases We can do this reasoning. For the first nut I have 4 possibilities: the numbers 1, 2, 3 and 4. Now, assume that the 1 comes out. In the second dice, the “eligible” numbers are 3 (2, 3 and 4), because if the second nut shows 1 we are no longer in the situation of the “blow of Venus”. For the third nut we only have 2 “eligible” numbers (the other 2 have already come out in the previous nuts) and for the fourth dice only 1 favorable number (the other 3 have already come out in previous dice). This applies to all 4 possibilities of the first nut. So we have: 4 cases favorable for the first nut; For each of these 4 possibilities we have 3 favorable cases for the second nut; For each of these 3 possibilities we have 2 favorable cases for the third nut; For each of these 2 possibilities we have only 1 possibility for the last nut. The total number of favorable cases is therefore:
favorable cases = 4 · 3 · 2 · 1 = 24
Ok, let’s move on to Possible cases. By making a similar reasoning we have: 4 possibilities for the first nut, 4 for the second, 4 for the third and 4 for the fourth. In total, therefore:
Total cases = 4 · 4 · 4 · 4 = 256
The probability of hitting the “blow of Venus” is the relationship between these two numbers. Dividing them we get:
Probability of Venus = 24/256 ≈ 0.09375
which corresponds to the 9.375%. Early unlikely, but not a lot: guessing a number of roulette (which has 37 numbers) has a probability of about 2.7%, therefore over 3 times less.
The question also asks for the probability of having 4 equal numbers. The reasoning is similar to what we have done above: the total cases are always 256, but the favorable ones are far less. Indeed, we can really count them one by one: it is the 1-1-1-1-1-2-2-2, 3-3-3 and 4-4-4-4 combinations. There are therefore 4 favorable cases. The probability is therefore:
probability 4 equal numbers = 4/256 = 0.0156
or the1.56% about.
