How often does trull occur on Whisthub?
When someone is dealt 3 or more aces in wiezen or colour whist, this game situation is called trull. The player in question is then obliged to announce this and is automatically paired with the player with the fourth ace (or in case of four aces, the player with ♥K etc.), and there are only a limited number of contracts that still go above trull. More information about this can be found in the rules of colour whist and wiezen.
Due to this forced nature, trull is not very popular with most players. Especially in colour whist, the bidding process of finding a suitable partner is one of the most fun aspects of the game, and trull breaks this, usually with a contract for 8 or 9 tricks that is easy to win. Because trull is so unpopular, color whist offers premium players the option to play without trull, so that even with three or more aces, bidding can still be done.
However, I regularly receive complaints that trull occurs much more often on Whisthub than in "real life", and the fact that premium players can turn off trull probably fuels this suspicion because it is supposedly a dark pattern to make more players pay for a subscription. Once and for all: this is not true. The cards on Whisthub are dealt as they would be dealt in real life.
To prove this, it's helpful to look at the probability of someone being dealt three or four aces. The math behind this isn't simple, so let's first look at the probability of you being dealt three or four aces if the cards were dealt completely randomly. For those who still know combinatorics from math classes, this can be calculated using binomial coefficients as
In other words, the probability that you will be dealt three or four aces is a good 4% in any game. This seems a lot less than the frequency of trull on Whisthub, and it is! After all, it is not only trull if you have three or more aces, but also when someone else has three or more aces! The 4% chance of trull should actually be multiplied by 4, because there are 4 players! In other words, the chance of trull per game is 16%, or once every 6 deals, much more in line with what players experience on Whisthub!
Real mathematicians will notice that this is not correct - and that is true. After all, the probabilities are not independent: if player A has three aces, then we know that no other player can also have three aces. So you cannot simply multiply the 4% chance by 4, but it is still a good approximation because the chance is small.
As mentioned, the exact calculation of the probability is much more complicated, but another way to calculate it is by means of simulations with a computer. The code for this in JavaScript is relatively simple:
// This function shuffles the cards completely randomly according to the
// Fisher-Yates algorithm (https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle)
function shuffle(cards) {
let m = cards.length;
while (m > 0) {
let i = Math.floor(Math.random() * m--);
let t = cards[m];
cards[m] = cards[i];
cards[i] = t;
}
}
// This function simulates a deal, counts how many aces each player has
// and then returns as a value whether it is a trull or not.
function isTrull() {
const cards = [];
for (let i = 1; i <= 13; i++) cards.push(i, i, i, i);
shuffle(cards);
const aces = [0, 0, 0, 0];
for (let player = 0; player < 4; player++) {
for (let i = 0; i < 13; i++) {
if (cards[13*player+i] === 1) {
aces[player]++;
}
}
}
return Math.max(...aces) >= 3;
}
// This function counts how many times trull occurs out of 1,000,000 deals, and
// returns that value as a percentage.
function simulate(n = 1e6) {
let counter = 0;
for (let i = 0; i < n; i++) {
counter += isTrull() ? 1 : 0;
}
return counter/n * 100;
}If we now execute simulate(), we get the probability that a game where the cards were dealt randomly is a trull game
or in other words, statistically 1 in 6 deals results in a trull, provided that the cards are dealt completely randomly! If you don't believe this, you can run the simulations below yourself. The counter indicates how many deals have already been simulated, along with the percentage of these deals that result in a trull. The counter has no maximum, so you can let the simulations run as long as you like. One thing is certain: the probability always converges to about 17.5%!
As explained in detail in this blog post, the cards are not just randomly dealt, but depend on the previous game. The probability of 17.5% as calculated above is therefore not correct. To determine how often trull really occurs on Whisthub, there is only 1 possibility, and that is to calculate the frequency based on all games that have ever been played on Whisthub.
On the date of writing, 6,605,638 games of colour whist have already been played on Whisthub in which trull was not turned off in the rules. Of these 6.6 million games, 844,149 were trull, i.e. 12.7%, or one in 8 games with trull. This means that the way the cards are dealt on Whisthub ensures that there is less trull than with completely random dealing!
Believe it or not, but those are the figures. However, even I sometimes have the impression too that trull occurs more often on Whisthub than in real life too, just like many players. How come? I don't have an immediate explanation for this, although I think the main reason is that the cards on Whisthub are dealt by the computer. This ensures that more games can be played per hour than in real life, because dealing in real life often takes a minute or two, especially if there is some discussion about the cards. When you play with friends in real life, it also happens that there is small talk, which makes the game go a bit slower too. In other words, the frequency of trulls is the same, but on Whisthub it feels higher because there are more trulls per hour!
This is of course just a guess. In any case, people are more suspicious of the cards when they do not see exactly how they are dealt (which is the case on Whisthub). That is more of a psychological aspect that can be probably fill many books.
The moral of the story, however, is this:
The number of trull games on Whisthub is not manipulated, and is even lower due to the way of dealing at 12.7% than if the cards were dealt completely randomly.