Pain, probability and the Puy: How bike racing works

Mike Woods won stage nine of the Tour de France atop the Puy de Dôme, after a complicated and absorbing battle between the 14 riders who made the break

The ‘problem of points’ was a maths problem initially posed by the Italian mathematician Luca Pacioli with reference to gambling.

We will get to the cycling.

Imagine a coin-flipping game between two people who have each staked £32, in which the first person to win a certain number of rounds wins the prize pot of £64. The problem Pacioli identified was how to fairly divide the pot if the game was stopped before finding its resolution. His solution was to look at the score, and divide the pot according to the number of rounds won. So a 10-5 lead would result in one player winning two thirds of the pot and the other player one third.

This sounds intuitively fair, but what if the game was interrupted after just one game, asked the 16th-century Italian mathematician Niccolò Tartaglia. One player would win 100 per cent of the pot, which in a large series, would be an unfair outcome. Tartaglia’s solution was to look at the number of rounds and the size of the lead and create a ratio based on these two numbers to divide the pot. However, in a game of 20 rounds, a five-point lead where one player had 19 points and the other 14 would probably be decisive, whereas the same score in a 100-round game would not be decisive. The problem was that Tartaglia’s solution led to the same division of the pot in each case.

Blaise Pascal, a mathematician, physicist and philosopher born in Clermont-Ferrand, finally solved the problem. Instead of looking at the results so far, he looked at the remainder of the game. How many rounds are there left? And what are the chances of each player winning enough rounds to win the pot? Imagine a game that is the best of seven, therefore the winner is the first to four rounds, and the pot is £64. The score is 2-1 to player A when the game is interrupted. Pacioli’s solution would divide the £64 pot by three, and give two thirds (£42.66) to player A and one third (£21.33) to player B. However, looking at the probability for the rest of the game, it is clear that what needs to happen for player A to win the pot is to get two wins before player B gets three, in the next four rounds. There are 16 possible outcomes to those four rounds, and 11 of those scenarios involve two wins for player A (the proof of this is at the bottom of this feature)*. Eleven out of 16 is a 68.75 per cent chance, and 68.75 per cent of the £64 pot is £44, which is £1.33 more than Pacioli’s method would calculate.

Read more: Agony and ecstasy on Puy de Dôme

Matteo Jorgenson, the Movistar rider who came within 400m of victory in stage nine of the 2023 Tour de France, which finished on the Puy de Dôme, might have wished that this game was stopped early. He’d put himself out in front of the 14-rider split which defined the day with 47km to go, and that group had split further in the American rider’s favour. He’d held the chasing quartet of Matej Mohorič, Neilson Powless, Mathieu Burgaudeau and David de la Cruz, before the last rider had a mechanical and dropped back, at a gap which slowly expanded to over a minute, and the rest of the group another 45 seconds behind. However, he was slowly chased down from that second group by the Canadian climber Mike Woods, who caught and passed the riders ahead of him – Powless, Mohorič and finally Jorgenson – in a relentless, implacable ticking off of names. Woods passed Jorgenson with 400 metres to go and won the stage.

Matteo Jorgenson

Matteo Jorgenson finished fourth on Puy de Dôme after being caught in the final kilometre (Zac Williams/SWPix)

Deciding cycling strategy is all about probability. You never know for sure what is going to happen; you can only ascribe it a probability. And then you sit and watch and hope. Jumbo-Visma tackled stage six of the 2023 Tour to Cauterets thinking that Tadej Pogačar was vulnerable, because he’d conceded a minute to their leader Jonas Vingegaard the day before, in a comparatively short distance. They rode in a way that aimed to exploit that high probability of vulnerability – putting Wout van Aert into the early break, then shredding the peloton on the Col du Tourmalet. The aim was for Vingegaard to drop Pogačar there, join up with Van Aert on the descent and valley road and put another chunk of time into their rival. However, Pogačar stayed with Vingegaard on the climb, and they could have reassessed their tactics then and sat up. But they might also have weighed up probabilities – the evidence of Pogačar being weaker the day before against the current evidence that he was going a bit better on this day – and still assumed that they were in the better position. Their mistake was the same as Luca Pacioli’s – at the moment of decision, don’t look at the result so far, but at the result which has not happened yet. Jumbo will no longer base their race on a high probability of Vingegaard being able to drop Pogačar. The fact that the lower probability outcome – Pogačar actually dropping Vingegaard – turned out to happen didn’t make Jumbo’s reasoning flawed, it just showed that sometimes your opponent turns up four heads in a row and it’s hard to do much about it.

Once it became clear that the 14-rider break on stage nine was going to stay away to contest the stage victory atop the Puy de Dôme, the riders within it had to work out their options, and weigh up probabilities. The important facts were that there were nine teams represented, and five of them – Astana, Israel-Premier Tech, TotalEnergies and Uno-X – had two riders. Clément Berthet (AG2R), Mohorič (Bahrain Victorious), Powless (EF-EasyPost), and Victor Campenaerts (Lotto Dstny) were on their own. Just as important: the presence of Woods in the break, with his team-mate Guillaume Boivin. Everybody would have known that Woods was the best climber, and therefore needed to be got shot of well before the finish; at the same time, they couldn’t begin the work of getting shot of Woods until the break itself was a safe distance in front of the peloton. Luckily, the peloton let the lead go out to well above 10 minutes, but for a long time, Wood’s biggest rivals for the stage win were also his biggest allies.

Pascal’s study of probability led him to conclude that it was a rational choice to believe in God, reducing the argument to an axiom known as Pascal’s wager. He suggested that God either exists or doesn’t, and that there is no way to know. And there is a choice to believe, or not. So there are four outcomes: one believes and God exists, which results in eternal bliss; one believes and God does not exist, which results in nothing; one does not believe and God exists, which results in eternal damnation; and one does not believe, and God does not exist, which results in nothing. Pascal argued that the only wise decision was to believe in God, since if you believe, you either gain everything or lose nothing, and if you do not, you either lose nothing or lose everything. Matteo Jorgenson’s wager with an hour and a bit of racing to go was equally rational: he could attack or not, and his rivals could chase, or not. The result of attacking was either a win, or loss. The result of sitting tight would be a loss. Therefore: attack. He wasn’t the first to come to this conclusion – the attacks started coming at 62km, when Boivin broke the truce. But the American’s attack at 47km, following at least six previous attempts by others, was the one that broke clear. The others didn’t chase, partly because after 15km of attacks and counterattacks, they were starting to look tired, but also because the presence of Jorgenson’s Movistar team-mate Gorka Izagirre in the group also dissuaded them. The Spanish rider is a good climber, and his rivals may not have been minded to allow him to get to the bottom of the Puy de Dôme at the front of the race, and fresher than everybody else, having sat in while the others chased Jorgenson.

Ten kilometres after Jorgensen’s attack, Mohorič went off in pursuit with Burgaudeau, followed soon after by Powless and De la Cruz. This put the American in a more vulnerable position – Izagirre was distanced, and so all four had good reason to work together. Furthermore, Burgaudeau and De la Cruz also had team-mates behind, which put a drag on the chase from the rest of the group. However, Jorgensen was maintaining a constant high pace, while Mohorič in particular kept surging and attacking, forcing the other three to chase. The peaks of speed with the chasing quartet were much higher than Jorgenson’s speed; however, the troughs were lower. What’s more, surging into the red zone again and again has a more erosive effect on endurance than that constant high pace.

Stage nine, Tour de France

The breakaway on stage nine of the 2023 Tour de France (James Startt/Agence Zoom)

In Clermont-Ferrand, Jorgenson led Mohorič, Powless and Burgaudeau by a minute, and the Woods group by 1:45. At five kilometres to go, just before the final steep ramp up the Puy, the gaps were 1:20 and 2:15.

Race followers at this point might still have concluded that the higher probability was that Jorgenson was going to win. He had spent more energy than most of his pursuers, but that was balanced out by his lead. Powless, on paper, was the best of the chasing trio at climbing, but Mohorič and Burgaudeau also knew this and made Powless chase into the wind up the climb out of Clermont-Ferrand, taking the sting out of his legs. Mohorič had the bigger engine, but was slower at climbing. Woods was the fastest climber of all, but a 2:15 gap at the bottom of a 20-minute climb looked too big to close.

However, the Canadian set about his task and patiently worked his way up the climb. For that final ascent of the Puy de Dôme Blaise Pascal would have warned Jorgenson: don’t look at what’s happened so far, look at what hasn’t happened yet.

* The 16 possible outcomes of four coin flips are: AAAA, AAAB, AABA, AABB, ABAA, ABAB, ABBA, ABBB, BAAA, BAAB, BABA, BABB, BBAA, BBAB, BBBA, BBBB. A has at least two wins in 11 of these outcomes. B has at least three wins in the other five.

Cover image by James Startt/Agence Zoom

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