Quick question which should be an easy one for all those who are "fluent" with such kind of calcs (I haven't done those in a while ...):

If we assume that, on average, say 15% of players cash in a tournament, how likely is it then, for instance, to cash 4 in a row (could have replaced 15 and 4 with "x" and "y", but perhaps it is less convoluted with actual numbers) ?

whenever someone asks "what's the probably of THAT", the usual trip-up is in defining THAT.

if an event has a 15% chance of occurring and each trial of that event is independent, then the probability of the event occurring over the next 2 trials is .15 * .15, and the probabilitity of that event occurring over the next 4 trials is .15 * .15 * .15 * .15 or 0.00050625 or .050625%

There is at least a few ways in which this is not the question you asked, but I'm gonna assume it's the answer you are looking for.

Wow, that was quick ... hmm, I was, at the time of writing, not aware that the question can be interpreted in several different ways, but you are right; I see now: this answer assumes that I play exactly 4 tournaments, and gives the probability of cashing all four of them.

The question - worded more precisely now, hopefully - was intended to mean how likely a streak of four (or more) cashes is, without a given number of tourneys.

E.g., the probability of getting "heads" with an infinite amount of coin tosses is 1/2. There will be, over that infinite anount of events, streaks of, say 10 or more times "heads" in a row, and there should be a mathematical probability of that happening, I suppose.

Regarding the tourney cash question then: how likely would a streak of 4 or more cashes "overall" be? Does the question, with this wording, make sense?

You're on the right track regarding the precision of the wording. When we ask the likelihood of 4 consecutive independent 15% outcomes over 4 trials, the answer is .15 * .15 * .15 * .15. BUT, if we ask how likely is it that a streak of 4 or more 15% outcomes will occur at least once over N trials, well that depends on how big N is, and is a more involved computation. (depends how many tournaments)

However, if 15% of the people cash in a tournament, that is only one piece of the puzzle in determining how likely it is for a given player to cash. In the real world, we don't know, we can only estimate that player's likelihood to cash.

If you had asked, "if we estimate that a given player has a 15% probability to cash in each tournament he plays, and each tournament result is independent, what is the probability that player will cash in all 4 of his next tournaments?" then the answer to that question is .15 * .15 * .15 * .15, or .050625%.

Are you sure that it is impossible to compute such a probability regardless of knowing N (i.e. N = infinite)?

Just as in the coin toss example: given a fair coin, a streak of e.g. 100 times "heads" in a row should be "very, very low" (but possible), and intuitively I would say that a probability of this ocurring should be calculable, no?!

(Phrased another way: let's say that millions of players play millions of poker tournaments each year, so that we may well assume a distribution of cashes close to that of an infinite amount of tourneys. How often then, percentage-wise, would players experience a 4-or-more streak of cashes when we assume equal skill for all players, i.e. the probability of cashing in every single tournament is 15% ?)

It's just 1 / .00050625

So on average it will take about 1975 mtts to get a streak of 4 cashes.

I ran 15million simulations and got 7665 such streaks, which is one in 1956.
If you only wanted to count distinct streaks, and not count a 5-cash-streak as two 4-cash-streaks, then I got one in 2309 mtts.

Lol, exactly 2 minutes ago I was beginning to write such a simulation myself, just seeing your answer right now, thanks

For 4 in a row out of n tournaments with 15% win probability, check out a streak calculator.

Thank you, I didn't know that such thing existed Seems like, for instance, with 1000 tournaments and a probability of cashing of 15%, the probability of a streak of 4 is 34.9%; for 10k trials it's 98.7%, which is way, way higher than Yoshi63's sim showed ... confused now

Thinking about it more: since we determined in the beginning that cashing 4 times in a row out of only 4 trials (which must be much less likely than a streak of 4 over many more trials) is ~0.05%, which is one out of 2000, it only makes sense that for many trials, the probability should go way up.

I didn't know such a thing readily existed either. I changed my simulation to run a bunch of 2k-long series, and got 5684 of 10,000 with at least 1 streak. This is pretty close to the 57.7% given by a streak calculator.

A quick google search revealed that this topic has been discussed on math.stackexchange. Here is a "closed" solution to the problem.

Before I started the thread, I tried for a bit to figure out the math by myself ... But wow, seeing this now, it seems like a really tough one for someone not specifically trained in this kind of stuff, I'm glad I asked

The probability can also be affected if the person is playing to maximize the chances of cashing at the expense of giving oneself a chance to go deep.

My lifetime cash rate is around 15-16% and the average weighted percentage of people cashing in the events I play in is probably around 11-12%.

I can almost guarantee you that if I was playing the tournament with the survival mind, I would easily increase my chances of cashing to 25% or so. Of course, that would make my tourn play -$EV

Absolutely. It was just meant as a mathematical question though, and the assumption was (implicitly) equal skill level and playing style etc., since otherwise it will kinda leave the mathematical realm, or at the very least become more complicated than it is already (e.g. percentage of players cashing will vary from one tournament to the next, so 15% is also just a rough number).

I need to get better at programming because I consider attempting a closed form solution to be easier than coding a simulation.