Expected gaps between events

If there is an event occurring, say 1 in 100 times for ease of use, how do I determine the chance of the gap between events happening? So basically a "games since last event" counter. How many times would you expect it to occur on successive games? How many with exactly a 1 game gap? How many with exactly a 2 game gap ? How many with a 1000 game gap? etc., I'm thinking an infinite number of samples, rather than a small set. Is there a statistical formula to use? Any ideas very much welcomed.

Probability of event is (1/100)
Probability of non-event is therefore (99/100)

P(gap size zero) is therefore [ (99/100)^0 ] (1/100)

P(gap size one) is therefore [ (99/100)^1 ] (1/100)

P(gap size nineteen) is therefore [ (99/100)^19 ] (1/100)

Sum of all probabilities of gaps from zero to infinity is 1
Obtained by infinite series summation


(Edited 7:23 pm PST 03 Feb)

Thanks. I was heading in that direction but had been diverted by Poisson theory which was a bit of a red herring!

Yes, Poisson is very useful in other circumstances but not here.

Quote: pwcrabb

Probability of event is (1/100)
Probability of non-event is therefore (99/100)

P(gap size one) is therefore [ (99/100)^1 ] (1/100)

P(gap size nineteen) is therefore [ (99/100)^19 ] (1/100)

Sum of all probabilities of gaps from zero to infinity is 1
Obtained by infinite series summation
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I was understanding the OP's question differently.

You say the P(1) = [ (99/100)^1 ] (1/100). This is the probability of one miss before a hit when you are starting at a hit.

In an infinite sequence, the hits come only once in 100 games. The probability of seeing a sequence of back to back hits starting at any particular game is 1 in 10,000. A sequence with two hits separated by exactly one miss appears with a probability 0.000099, etc.

Perhaps the OP can clarify which number he is seeking.

Quote: BasilTheBorder

If there is an event occurring, say 1 in 100 times for ease of use, how do I determine the chance of the gap between events happening? So basically a "games since last event" counter. How many times would you expect it to occur on successive games? How many with exactly a 1 game gap? How many with exactly a 2 game gap ? How many with a 1000 game gap? etc., I'm thinking an infinite number of samples, rather than a small set. Is there a statistical formula to use? Any ideas very much welcomed.
link to original post

"how do I determine the chance of the gap between events happening?" leads me to believe it was my interpretation.

If you start out afresh and ask what are the chances of an event with pr=1/100 happening on the 1st, 2nd etc. attempt; then the answer is 1st = 1/100, 2nd = 99/100 * 1/100 etc. This gives a distribution of "gaps" which just gradually decreases as the gap gets larger.

By pure coincidence there's a similar discussion using craps dice asking how often would it take to throw a "2" (1-1) and then mentions other totals including a "Natural Winnerl" (7 or 11). The enclosed picture (I hope ChumpChange doesn't mind me copying it here) shows how the frequencies decrease. In this case there are 8 ways out of 36 to throw a "Natural Winner". I've since replied by comparing this to an example of a long road where some of the buildings are pubs and the rest private houses, asking how far is it to the next pub. https://wizardofvegas.com/forum/gambling/craps/37930-roll-frequencies-after-69-6k-rolls-on-wincraps/#post880610

Quote: ChumpChange

link to original post

I started my thread after reading the OP's post of this thread. It wasn't quite coincidence, but I didn't want to clog up the OP's thread. The replies here are beyond my typical math abilities.

Quote: charliepatrick

If you start out afresh and ask what are the chances of an event with pr=1/100 happening on the 1st, 2nd etc. attempt; then the answer is 1st = 1/100, 2nd = 99/100 * 1/100 etc. This gives a distribution of "gaps" which just gradually decreases as the gap gets larger.

OP specifically asked about "gap between events" but maybe OP meant gaps before the first event. These two are certainly not the same question.

CharliePatrick and I do not disagree. His pub scenario can be modeled similarly to Basil's scenario, except with differing probabilities for P(success) and P(failure). Mental requests from Basil a clarification and a specific problem, and I agree. Precise problem specification is essential.

Basil, are you analyzing for:
1. The probability of first success on exactly a specific future trial?
2. The cumulative probability of first success on or before a specific future trial?
3. The expected number of gaps, or consecutive failures, of a certain size within a large universe of experiments?
4. The expected size of the next gap?
5. The mean size of all gaps?
6. Something else?

What I am interested in is the profile of gaps between successful events. Disregarding the lead up to the initial event, I think Mental's explanation fits my scenario. I have a slot game in development with a random event occurring 1 in 100 games. Logging the gaps between events gives a similar shaped curve to using Mentals approach. For some reason I was expecting a bell shaped curve centred around the 100 game gap. I need to brush up on my statistical knowledge! So in answer to your question, 6, or possibly 3. The profile of gaps between successful events.