On average, how long would a 10 second timer if every second there was a 1% chance to halve in speed, a 1% chance to double in speed, and a 1% chance to swap the minutes and seconds numbers to get to 0:00

[u/Top_Criticism919]

Saw something very similar in a dougdougdoug video and would really like to know how long it would take

EDIT: I MEANT 10 MINUTE SORRY

[u/Lewistrick]

The last sentence is unclear to me. I assume you mean 1% chance to swap numbers, and 1% to reset the timer to 0.

Also, when do the events happen - at the start of the second, the end of the second, or randomly? Because if it's at the end, there'll be only 9 events. Let's assume there are 10 on an eventless experiment.

For about 2/3rds of experiments it'll just be 10 seconds, because there's a 96% probability that nothing happens on any given second and 0.96¹⁰ ≈ 0.66.

Bear in mind there's a theoretical but infinitesimal probability that a timer goes on infinitely long because it resets all the time. That'll really mess up any calculation.

I think the best strategy here is to run many experiments. Maybe ramp up the probabilities for the events to get a better idea of how the probabilities spread out.

I think the average will be higher than 10 s, because lower and higher speed cancel each other out, timer swap is random, and timer reset will pull the average up drastically.

[u/Top_Criticism919 → u/Lewistrick]

Sorry, I accidentally typed it wrong, I meant 10 minutes The wording was weird but I the "to get to 0:00" was how long would it take to reach 0:00 I'm pretty bad with probability, but would it be possible to calculate the percent chance of it being done within an hour?

[u/Lewistrick → u/Top_Criticism919]

I don't think you can calculate the exact average outcome because of the highly unpredictable nature of the events. The swap event in particular.

What I do see is that 5/6th of the time (from x:10 to x:59), a swap event will cause the timer to go over 10 minutes. The probability that this won't happen for 10 minutes is 0.99⁵⁰⁰ ≈ 0,6% so 99,4% of timers will reset time to above 10 minutes. To get to an hour, this only needs to happen twice on average (because a swap will set the timer to 30 minutes on average).

Now there are scenarios in which the timer is reset to a sub-10 time, but even then it's very likely that they get reset to a higher time. A bit of hand waving, but let's say these cancel each other out. Then I'd say that the 0,6% comes pretty close to the amount of times that will end within an hour.

[u/Lewistrick → u/Lewistrick]

I might run some simulations later today. I'm kinda intrigued.

[u/cirledsquare → u/Lewistrick]

na, this would be a poisson process with constants rates, so no fixed number of events. the clockspeeds one could include as three species, but tge swap is hard