Probabitity question for the math guys/gals, non-BlackJack related

[Nikky_Flash]

Something kinda odd happened to me the other day, and it seemed way high off the probability charts.
I sometimes work nights and I was anxious to leave , so I looked at the clock a few times , I looked and it said 6:44 , I thought to myself "I just told my wife the other day that I keep looking at digital-clocks when the minutes are :44 , and I told her it didn't seem mathematically likely to see it like half the times somedays, and other days about 1/3 of the time. but I didn't want to be seem like I was going a little batty - so I decided to forget about it"

now at work I did my job without looking at the clock since the 6:44 glace / until it seemed a decent amount of time went by and wanted to see how close I was to being able to leave and lo and behold again ;
it was 7:44 pm. now I didn't peek at a clock inbetween these times , and
next time I looked near closing time it was 8:44 pm . what was happening during the last few weeks of seeing this occur I was really thinking I was seeing that WAY more than mathematically probable ; sure I could've chalked up to overactive imagination, but three times in a row now seems too weird. So I applied math to the numbers, just to make sure: now here's where I can make a mistake as an amateur but
(correct me if I'm wrong, please ) but,
the chances of the clock being on any given number at any time you look seems to be 1.6666~% . Using that as a rough guide ,
I think I did it right when adding into a scientific calculator the odds of that happening three times in a row by using
.016667 to the power of three (?)
that equals the number 0.000004632407963
does that mean the changes of me seeing that 3 times in a row is; 1 in 4,632,407 ?
and if that's correct , how would I calculate it happening about half the time in a given day when i've looked 20 times / or a third of the time in a given day when looked 15 times or something... i don't know the formula for that; would be nice to look at ,
I post this question here because you all are the best at math and probability that I know...I don't want to obsess about this , or spend too much time since I try to put my math-free time into BlackJAck practice, and/or my other side job , but I figured it's a good way to learn more math too; and I could finally answer my own question; web searches didn't have too much on my exact situation , and you guys as a group tackle these things better than most websites do in my opinion ; thanks , Nikky

[RS]

My guess is selective memory.

[Freightman]

Something kinda odd happened to me the other day, and it seemed way high off the probability charts.
I sometimes work nights and I was anxious to leave , so I looked at the clock a few times , I looked and it said 6:44 , I thought to myself "I just told my wife the other day that I keep looking at digital-clocks when the minutes are :44 , and I told her it didn't seem mathematically likely to see it like half the times somedays, and other days about 1/3 of the time. but I didn't want to be seem like I was going a little batty - so I decided to forget about it"

now at work I did my job without looking at the clock since the 6:44 glace / until it seemed a decent amount of time went by and wanted to see how close I was to being able to leave and lo and behold again ;
it was 7:44 pm. now I didn't peek at a clock inbetween these times , and
next time I looked near closing time it was 8:44 pm . what was happening during the last few weeks of seeing this occur I was really thinking I was seeing that WAY more than mathematically probable ; sure I could've chalked up to overactive imagination, but three times in a row now seems too weird. So I applied math to the numbers, just to make sure: now here's where I can make a mistake as an amateur but
(correct me if I'm wrong, please ) but,
the chances of the clock being on any given number at any time you look seems to be 1.6666~% . Using that as a rough guide ,
I think I did it right when adding into a scientific calculator the odds of that happening three times in a row by using
.016667 to the power of three (?)
that equals the number 0.000004632407963
does that mean the changes of me seeing that 3 times in a row is; 1 in 4,632,407 ?
and if that's correct , how would I calculate it happening about half the time in a given day when i've looked 20 times / or a third of the time in a given day when looked 15 times or something... i don't know the formula for that; would be nice to look at ,
I post this question here because you all are the best at math and probability that I know...I don't want to obsess about this , or spend too much time since I try to put my math-free time into BlackJAck practice, and/or my other side job , but I figured it's a good way to learn more math too; and I could finally answer my own question; web searches didn't have too much on my exact situation , and you guys as a group tackle these things better than most websites do in my opinion ; thanks , Nikky

I got up for a few minutes, noting the clock said 1.15 and promptly went back to bed. A little later, got up to pee, noting the time was 2.15.
Self, I asked - what is the probability of looking at the clock a second time Exactly 1 hour after checking it out the first time. Well, the first one is obvious in that there are 60 minutes in 1 hour, so the odds if any given minute being picked, is therefore 69-1 (must be thinking about something else) - make that 60-1. Checking the clock again, with the minute indicator being exactly the same us another 60-1 shot - so - the odds of choosing the same minute on 2 consecutive hours is 60x60 divided by 1x1. Looks like a 3600-1 shot. Incredibly, got an itch exactly one hour later and noted the clock to be exactly 3.15. So, what are the odds of picking exactly the same minute on 3 consecutive hours - that would be the answer above if 3600-1 multiplied by another 60-1, or in other words, 216000 -1.
It's been along night, and ironically, exactly 60 minutes later, I gaze at the click noting the time to be 4.15. Gave the wife a nudge - she told me to drop dead - it's the middle of the night. Oh - that 4th occurrence is 12960000 to one - and Dk it goes.

[Bodarc]

1 in 306,332 I believe for 1/3 of the time out of 15 looks at the clock

https://en.wikipedia.org/wiki/Bernoulli_trial

Probability of p(success) = 1/60 since 44 is one minute out of 60 minutes
Probability of q(failure ) = 59/60 since any number other than 44 is a failure
k=5 n=15

where is is a Binomial Coefficient solved by

We'll get the correct answer when Don wakes up!

In any case, I think I'd take my clock to a repair shop.

[Three]

(correct me if I'm wrong, please ) but,
the chances of the clock being on any given number at any time you look seems to be 1.6666~% . Using that as a rough guide ,
I think I did it right when adding into a scientific calculator the odds of that happening three times in a row by using
.016667 to the power of three (?)
that equals the number 0.000004632407963
does that mean the changes of me seeing that 3 times in a row is; 1 in 4,632,407 ?
a

If each look has the same chance of being any number (something that is not the case here) you have a 1/60 chance of seeing the number each time. You have a 1/(60*60*60) chance of seeing your number 3 times or 1/216000 (1 in 216,000). You got the decimal right but your way of on the ratio.

[RS]

The chance of seeing the same number 3 times in a row is 1/(60*60) or 1/3600. The chance of seeing the first number is 1 (certain).

The chance of seeing a specific number 3 times in a row is 1/(60*60*60), since the chance of seeing the first, specific number, is 1/60.


In other words, if you say "what's the chance that the first number I see...I will see it two more times on my next 2 looks?", that's going to be 1/(60*60).

If you say, "what's the chance that the first number I see is 23, and the next two times I look it will also be 23?", then that's going to be 1/(60*60*60).

[Bodarc]

Yes but the problem presented is to look at it 15 times and see it 5 out of the 15. You are working a different problem.

[DSchles]

OK, a few comments (I woke up! :-)).

Most of what's written above seems fine to me. For example, you really shouldn't count seeing any specific time three times in a row as raising that probability (1/60) to the third power, because, as stated above, the first time has to be something. So, unless you were specifically aiming for the 44, the coincidences begin to occur only with the second sighting.

The binomial formula is also correct. Let's use easier numbers. Suppose something happens 1/3 of the time and, therefore, doesn't happen 2/3 of the time. You get 10 chances, and you want to know the probability that the event will happen, say, six times. One such way that this could occur would be for it to happen the first six times in a row (1/3)^6 and then not happen the next four (2/3)^4. The answer doesn't matter, here, just the methodology. But, now, you have to understand that six successes followed by four failures is just one of the very many ways you could "place" the six successes in the ten slots. In all, there are 10C6 (ten choose six) ways of doing this, and so the original fraction must be multiplied by this value, making the overall probability, of course, much lower than the original calculation.

Go here for the perfect application to do this kind of work: http://vassarstats.net/binomialX.html

Note that the answer to the above would be 5.7% (n = 10, k = 6, p = 1/3).

Finally, allow me one more comment. If you are noticing a time during the day, or a time when you wake up in the middle of the night, all the minutes are certainly not equally likely. Given that your sleep cycle might have you waking in approximately one-hour increments, if you started at, say 44 minutes, you would be more likely than just 1/60 to hit 44 upon the next awakening. Those values only 10, 20, 30, or even 40 minutes higher would not have equal status with the minutes nearer to 44. The same may be true simply for looking at the clock during your job. You might just sense when about an hour has passed and take a look. As such, those minutes in the ranges above would be less than 1/60 likely and those closest to the hour mark somewhat more so.

Hope this helps.

Don

[Bodarc]

"I think I did it right when adding into a scientific calculator the odds of that happening three times in a row by using
.016667 to the power of three (?)
that equals the number 0.000004632407963
does that mean the changes of me seeing that 3 times in a row is; 1 in 4,632,407 ?"

To answer the other part of your question, I think it would be 1/.00000463247963 or

1 in 215,867

As was pointed out, that is if you were looking for a certain number and not taking the first number you saw.

[DSchles]

Actually, if you don't use any rounding, but just the fraction 1/60, the answer is, quite precisely, 1 in 216,000.

Don

[Bodarc]

See!!!! I told you what was going to happen when he woke up!

[Nikky_Flash]

thanks guys .. real good answers ... and I believe selective memory is likely involved ... and Dons idea of certain timing ..also learned the correct formulas.. I will write more when I'm free ... thanks

[Three]

But for months I'd turned in the middle of the night to look at the digital clock by my bed and it always read 3.16. I didn't understand the significance of this coincidence until months later.

Okay John, isn't that the number that those religious nuts that warn people that the end is near? Nope, I looked it up and that really isn't applicable to those religious types.