Probability of a streak

The main accusation of Kramnik relates to the mathematical unlikeliness of having a series of consecutive wins, but no math is presented, so the arguments are literally of no value. I will try to elevate the debate and present some math, which I'm sure chess.com & Ken Regan have already done (disclaimer: I did not proof check this with pair review, so community, feel free to correct me and carry on my research).

We are trying to find the probability of at least k consecutive successes in a Bernoulli trial (random independent experiments with 2 outcomes, success or failure). This is actually likely slightly incorrect given that there is a dependance between games, due to fatigue, psychological distress / motivation of being in a losing / winning streak, gained knowledge of the opponent's strengths & weaknesses (play style, openings, ...), amongst other factors, but we'll ignore this for the sake of simplicity (and intuition leads to believe it would favor the stronger player).

We set n independents trials of the same event E which has a individual probability p of success, and we want the probability pr(n, p, k) to have at least k successes in a row.

This website gives us a recursive formula to calculate pr(n, p, k), which translates to the following python code, after memoization:

If you want to do some experimentations yourself, you can use this link, change the values of n,p,k and click "run". It will output the probability of having at least k consecutive wins in a series of n games with a probability p of winning against your opponent.

Now for the usecase at hand and the accusation, I will use numbers already provided, but feel free to change those in the python program to get other results:

• Number of games: 50000, of which 40% are to be disgarded (see Vladimir's post : "Which immediately decrease the amount of games by a good 40 percent."), so n = 30000 (incidentally, it also roughly matches the number of blitz games Hikaru has played, as shown here)
• Number of consecutive wins: 45 ("45.5 out of 46"), so k = 45
• Probability of winning: I will take a guess here and assume Hikaru is on average 3200, his opponents during the streak were on average 2800 (the average rating of his blitz opponents is currently 2793 as shown here) so 400 points lower, so it gives us a probability of 90.91% to win (see glicko formula here). In its original message that started all of this (here), Vladimir said the average opponents were 2950, and though I don't know what Hikaru's rating was, you can use a 300 point elo difference and check for yourself, you'll see the result is similar. The actual probability should be higher because Hikaru can choose his opponents and will stop playing if he feels he can't win easily (in other words, he will only play overrated players), but I won't consider that. So let's use p = 0.9 (90%)

And finally for the conclusion and interesting part, using those numbers and the formula above:

Hikaru has a 99.99999999985469% chance of having at least a 45 winning streak in 30k games !!!

So I guess chess.com were right when they said "We have found that not only is a 45 game winning streak possible, it is in fact likely given the number of games played", contrary to what Vladimir says in this post. It is in fact ALMOST MATHEMATICALLY IMPOSSIBLE (0.00000000014531 %, or a chance of 1 in a trillion) not to have such a streak.

Side note, Kramnik's claim now focuses on the improbability of the number of streaks, rather than their existence (though he stills refutes chess.com's statement, which only mentions the existence of such streaks), but regardless, the math shown above makes it clear such streaks happen quite frequently, even naturally, and Hikaru is known to seek a pick rating by playing against much lower rated players, as opposed to other top GMs.