# 20 independent events happening in 10 hour period Probability breakdown for at least 2 in an hour? **URL:** \ **Category:** Factual Questions\ **Created:** [October 2, 2024, 8:42pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388 "2024-10-02T20:42:19Z")\ **Posts on this page:** 19\ **Page:** 1
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [October 2, 2024, 8:42pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/1 "2024-10-02T20:42:19Z")
Could somebody calculate this for me? Over a 10 hour period event X will happen exactly 20 times, independently random across the 10 hour period. If we split the 10 hour period up into 10 discrete hours and check them individually is there any way to calculate the probabilities for how many of them have at least 2 events? ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [October 2, 2024, 8:50pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/2 "2024-10-02T20:50:52Z")
Am I correct that this equivalent to saying “roll a d10 twenty times; what is the probability that a particular number 0-9 is rolled at least twice”? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [October 2, 2024, 8:56pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/3 "2024-10-02T20:56:31Z")
I think it’s how many of the numbers came up twice or more. ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [October 2, 2024, 9:03pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/4 "2024-10-02T21:03:44Z")
> [@Andy\_L](#): > > I think it’s how many of the numbers came up twice or more. Yes that would be exactly equivalent. ---
**Author:** ![Snarky\_Kong](https://avatars.discourse-cdn.com/v4/letter/s/a183cd/32.png) [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)\ **Post date:** [October 2, 2024, 9:04pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/5 "2024-10-02T21:04:04Z")
I’m getting: - 0: 0% - 1: 0% - 2: 0% - 3:0.3% - 4: 4.3% - 5: 23% - 6: 39.6% - 7: 26% - 8: 6.5% - 9: 0.4% - 10: 0% eta: low precision answer. ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [October 3, 2024, 10:45am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/6 "2024-10-03T10:45:46Z")
Ok thanks. ---
**Author:** ![Gordon\_G](https://avatars.discourse-cdn.com/v4/letter/g/3da27b/32.png) [@Gordon\_G](https://boards.straightdope.com/u/Gordon_G)\ **Post date:** [October 3, 2024, 3:54pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/7 "2024-10-03T15:54:15Z")
I don’t know a good way to get the answers except by random simulations. My results are in the same ballpark as those of Snarky\_Kong. - 0: 0% - 1: 0% - 2: 0.003% - 3:0.257% - 4: 4.37% - 5: 22.2% - 6: 40.0% - 7: 26.7% - 8: 6.09% - 9: 0.368% - 10: 0.002% ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [October 3, 2024, 4:03pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/8 "2024-10-03T16:03:44Z")
How many simulations? ---
**Author:** ![Gordon\_G](https://avatars.discourse-cdn.com/v4/letter/g/3da27b/32.png) [@Gordon\_G](https://boards.straightdope.com/u/Gordon_G)\ **Post date:** [October 3, 2024, 4:37pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/9 "2024-10-03T16:37:12Z")
300 million! (Computers are fast these days.) I didn’t calculate error-bars. I did 3 runs of 100 million each and eye-balled the variation; there seemed to be slightly more sig-figs than I showed. ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [October 3, 2024, 4:37pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/10 "2024-10-03T16:37:46Z")
Thanks a lot. That should make the results as good as they can be. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 4, 2024, 5:32am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/11 "2024-10-04T05:32:06Z")
> [@Gordon\_G](#): > > I don’t know a good way to get the answers except by random simulations. I just used a computer algebra system. You don’t need to simulate anything, just symbolic algebra. 0: 0.000% (i.e. zero) 1: 0.000% (i.e. 0.0000000123320412181) 2: 0.003% 3: 0.257% 4: 4.371% 5: 22.2% 6: 40.0% 7: 26.7% 8: 6.09% 9: 0.368% 10: 0.002% ---
**Author:** ![Moonrise](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/moonrise/32/14764_2.png) [@Moonrise](https://boards.straightdope.com/u/Moonrise)\ **Post date:** [October 4, 2024, 7:16am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/12 "2024-10-04T07:16:43Z")
Do these results mean that there’s a 40% chance that 6 one-hour time slots have at least 2 events? (I’m not a mathematician) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 4, 2024, 7:41am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/13 "2024-10-04T07:41:36Z")
That’s for the op to say, but that is how I interpreted it: you have 20 balls, and you throw them randomly into 10 bins. Then you count what is the probability that exactly 6 bins received at least 2 balls, you get 40% ---
**Author:** ![Moonrise](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/moonrise/32/14764_2.png) [@Moonrise](https://boards.straightdope.com/u/Moonrise)\ **Post date:** [October 4, 2024, 8:21am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/14 "2024-10-04T08:21:03Z")
Thanks. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [October 4, 2024, 11:05pm UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/15 "2024-10-04T23:05:30Z")
I did some simulations too (a bit over 4 billion of them) 1 hour with more than 2 events occur 0.000001247% of the time 2 hours occur 0.00283% of the time 3 hours occur 0.2565% of the time 4 hours occur 4.37% of the time 5 hours occur 22.2% of the time 6 hours occur 40.03% of the time 7 hours occur 26.67% of the time 8 hours occur 6.1 % of the time 9 hours occur 0.368% of the time 10 hours occur 0.00238 % of the time Further thoughts: There at 10^20 possible outcomes. For the case of only 1 hour having more than 2 events, there are 10 ways for that to happen with all the events in one hour 90 ways (10 times 9 choose 1) for that to happen with 19 events in one hour and 1 in a different hour 360 ways (10 times 9 choose 2) for that to happen with 18 events in one hour, and two in two different slots in the 9 hours remaining and so on. That gives a total of 5120 possible ways to end up with only one hour having more than 2 events (out of 10^20) which suggests my simulation is not doing an accurate job with the very unlikely events, which I can believe. But working through the problem this way, makes me think we’re dealing with something very much like entropy. The only-one-hour-with-2-or-more-events macrostate is very unlikely, and a small change (moving one event) probably changes the macrostate. The most likely case is the lowest entropy, with small changes probably leaving you in the same macrostate. What do our physicist friends think @Chronos and @Asympotically_fat ? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 5, 2024, 2:58am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/16 "2024-10-05T02:58:38Z")
> [@Andy\_L](#): > > 10 ways for that to happen with all the events in one hour OK… > [@Andy\_L](#): > > 90 ways (10 times 9 choose 1) for that to happen with 19 events in one hour and 1 in a different hour What happened to the 20 events? You mean 90 times 20 > [@Andy\_L](#): > > That gives a total of 5120 possible ways to end up with only one hour having more than 2 events (out of 10^20) We had 1233204121810 ---
**Author:** ![Gordon\_G](https://avatars.discourse-cdn.com/v4/letter/g/3da27b/32.png) [@Gordon\_G](https://boards.straightdope.com/u/Gordon_G)\ **Post date:** [October 5, 2024, 7:19am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/17 "2024-10-05T07:19:37Z")
> [@DPRK](#): > > I just used a computer algebra system. You don’t need to simulate anything, just symbolic algebra. I took this as a challenge and wrote a simple procedure that gives _exact_ results. The procedure can be translated into cumbersome math formulae. > [@DPRK](#): > > Then you count what is the probability that exactly 6 bins received at least 2 balls, you get 40% 40.032742724298324 % to be precise; or if you prefer, 10008185681074581 / 25000000000000000 ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 5, 2024, 7:37am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/18 "2024-10-05T07:37:40Z")
I am not being coy (though I agree it is a fun challenge). If anyone wants, I am happy to post some cumbersome math formulae. In fact, here you go: > **Summary** > > We will use the interpretation in Post #2, or we can think of the periods as labelled `A` through `J` and form random 20-letter words like `IFHJGDJFCFFDEAEIFHDI`. > > Now we want [the computer] to count how many letters show up at least twice. A quick way is, define an exponential generating function and introduce a variable u to mark the events of interest: > > \bigl(1+t+u(\exp(t)-t-1)\bigr)^{10} > > Finally, read off the counts by taking the coefficient of t^20 and multiplying by 20!: > > 2375880867360000 \, u^{10} + 368261534440800000 \, u^{9} + 6094662398304480000 \, u^{8} + 26671998421164096000 \, u^{7} + 40032742724298324000 \, u^{6} + 22199535666452944800 \, u^{5} + 4371032227712729040 \, u^{4} + 256556472386987520 \, u^{3} + 2833441168156830 \, u^{2} + 1233204121810 \, u ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [October 5, 2024, 10:27am UTC](https://boards.straightdope.com/t/20-independent-events-happening-in-10-hour-period-probability-breakdown-for-at-least-2-in-an-hour/1008388/19 "2024-10-05T10:27:47Z")
> [@DPRK](#): > > What happened to the 20 events? You mean 90 times 20 You’re right. I got stuck on the number of slots and not the number of events and got the wrong multiplication. Embarrassing. > [@DPRK](#): > > We had 1233204121810 Which is actually very close to my simulation. So that’s cool.