double digits: what's the probability of seeing double digits on a clock?

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Girl##

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apologies in advance for my stupidity - this probability poser has been bothering me for a while..
here it is: what's the probability of seeing double digits on a clock? in an average day, there's a chance of me seeing 17 double-digits on my phone when i glance at it. the first is at 07:07 and the last is 23:23. i have no idea where to start to work this out - is it how many seconds in in a 16hr day divided by 17? i have no clue! but i'm hoping someone out there has..
i see double digits up to 3 times a day and i look at my phone maybe up to 10 times during the day. so what are the chances? and no, i don't believe in angel numbers or any other mystical explanations.
 
Girl## said:
apologies in advance for my stupidity - this probability poser has been bothering me for a while..
here it is: what's the probability of seeing double digits on a clock? in an average day, there's a chance of me seeing 17 double-digits on my phone when i glance at it. the first is at 07:07 and the last is 23:23. i have no idea where to start to work this out - is it how many seconds in in a 16hr day divided by 17? i have no clue! but i'm hoping someone out there has..
i see double digits up to 3 times a day and i look at my phone maybe up to 10 times during the day. so what are the chances? and no, i don't believe in angel numbers or any other mystical explanations.
Click to expand...
First, we need to be sure of the assumptions. It appears that you are assuming you only see a clock from 07:00 to 23:59. Is that what you mean? (It isn't true of everyone!) You have to clearly say so in order to define your problem. Also, clearly, you are talking about a 24-hour clock; presumably it shows only the hour and minute, in that format. (So you don't need to count seconds.)

But also, it appears that your definition of "double digit" means that (at least?) two of the digits shown are the same (not necessarily consecutive), and that double 0 doesn't count. Or maybe, based on your examples, you only mean that the hour and minute are the same. There may be other issues as well.

Now, assuming that the problem is as I have just stated, you need to first count the number of different times you see in a day (that is, how many minutes there are in this interval). Then you need to work out how many of those times contain two of the same digit. That's not hard, but also not trivial. One way to approach it might be to find an orderly way to count the number of occurrences of two or more 1's, and so on. (But then you'd have to consider that there may be times when two different digits appear twice, and subtract that count.)

Give it a try. (You could also make a spreadsheet of times, and use that to do your count.)

As to your surprise at the apparent frequency of this happening, you may want to actually keep a count in one day of the number of times you see a clock, and the number of times there is a double digit. You may well find that you are undercounting the former, and noticing only the days when the latter is unexpectedly large. (You mention the numbers 17 and 3 for this; where do they come from?)
 
Dr.Peterson said:
First, we need to be sure of the assumptions. It appears that you are assuming you only see a clock from 07:00 to 23:59. Is that what you mean? (It isn't true of everyone!) You have to clearly say so in order to define your problem. Also, clearly, you are talking about a 24-hour clock; presumably it shows only the hour and minute, in that format. (So you don't need to count seconds.)

But also, it appears that your definition of "double digit" means that (at least?) two of the digits shown are the same (not necessarily consecutive), and that double 0 doesn't count. Or maybe, based on your examples, you only mean that the hour and minute are the same. There may be other issues as well.

Now, assuming that the problem is as I have just stated, you need to first count the number of different times you see in a day (that is, how many minutes there are in this interval). Then you need to work out how many of those times contain two of the same digit. That's not hard, but also not trivial. One way to approach it might be to find an orderly way to count the number of occurrences of two or more 1's, and so on. (But then you'd have to consider that there may be times when two different digits appear twice, and subtract that count.)

Give it a try. (You could also make a spreadsheet of times, and use that to do your count.)

As to your surprise at the apparent frequency of this happening, you may want to actually keep a count in one day of the number of times you see a clock, and the number of times there is a double digit. You may well find that you are undercounting the former, and noticing only the days when the latter is unexpectedly large. (You mention the numbers 17 and 3 for this; where do they come from?)
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Hi Maths Peoples..
I'll try to clarify, sorry.

I wake up at 7am and go to sleep before midnight so there are only ever 17 ocassions during the course of one day when I can see double-digits on my phone's 24hr clock.
The first is at 07:07, then 08:08, 09:09, 10:10, 11:11, 12:12, 13:13, 14:14, 15:15, 16:16, 17:17, 18:18, 19:19, 20:20, 21:21, 22:22, 23:23 = 17.
I record how many times I see double-digits and it's every day, 1 to 3 times a day.
I don't often look at the clock during the day, maybe ten times at most.

So.. if there are 16hrs in my day times 60mins in an hour = 960 minutes in my day when I could be looking at my clock.
How do I work out the probability of seeing double-digits?

Maybe you could point me in the direction of where I could go online for someone to answer this question for me?

Thank you :)
 
Girl## said:
apologies in advance for my stupidity - this probability poser has been bothering me for a while..
here it is: what's the probability of seeing double digits on a clock? in an average day, there's a chance of me seeing 17 double-digits on my phone when i glance at it. the first is at 07:07 and the last is 23:23. i have no idea where to start to work this out - is it how many seconds in in a 16hr day divided by 17? i have no clue! but i'm hoping someone out there has..
i see double digits up to 3 times a day and i look at my phone maybe up to 10 times during the day. so what are the chances? and no, i don't believe in angel numbers or any other mystical explanations.
Click to expand...
Looking at this again, I see I've drastically over-interpreted it. From the number 17, I see that you don't really mean double digits at all -- just, as I said in passing,
Dr.Peterson said:
Or maybe, based on your examples, you only mean that the hour and minute are the same.
Click to expand...
So you're merely saying that there is one time each hour when this happens, namely

07:07​
08:08​
09:09​
10:10​
...​
23:23​

If you look at a clock at a random time, then you will see this 1/60 of the time -- in any hour there are 60 different things you might see, and one of them is this type. That's all there is to it. And the same would be true if you wake up and go to sleep at the end of any hour.

To put it another way, in one waking day there are 17*60=1020 minutes; out of those, 17 will display in the form you are asking about; so the probability is

number of successes / number of tries = 17/1020 = 1/60 = 0.0167 = 1.67%​

If you looked 10 times and saw it 3 times, that would be a probability of 3/10 = 0.03 = 30%. It's almost certain that you are underestimating the number of times you look at a clock in a day.

The first step in solving a problem is to state it clearly. That's why we have carefully-defined words like "digit", which means just one of the symbols used in writing a number. Use the wrong word, and people will misunderstand you.
 
Girl## said:
I wake up at 7am and go to sleep before midnight so there are only ever 17 ocassions during the course of one day when I can see double-digits on my phone's 24hr clock.
The first is at 07:07, then 08:08, 09:09, 10:10, 11:11, 12:12, 13:13, 14:14, 15:15, 16:16, 17:17, 18:18, 19:19, 20:20, 21:21, 22:22, 23:23 = 17.
I record how many times I see double-digits and it's every day, 1 to 3 times a day.
I don't often look at the clock during the day, maybe ten times at most.

So.. if there are 16hrs in my day times 60mins in an hour = 960 minutes in my day when I could be looking at my clock.
How do I work out the probability of seeing double-digits?
Click to expand...
I've already answered this, after you wrote, due to the delay in approval (sorry about that). Your response wasn't visible until two hours ago.

Can you explain, first, why you call this "double digits" (rather than, say, "double numbers"), and, second, why you have changed the number from 17 to 16 here? The list is very helpful in clarifying the meaning, but the wording should have been updated based on what I said the first time.

I'll suggest, again, that you should record not only the times you see "double numbers", but also the actual number of times you look at the clock. It's very easy to notice only the times you seem something interesting, and forget all the times you didn't. If some of those are long looks (say, taking 10 minutes), that could also explain what you're seeing.

But in any case, probability is calculated by dividing the number of times something happened, by the number of times you tried, which here is 17/1020, or 1/60.
 
The times that you mentioned are always in the 1st half of an hour. Is there any reason why you might look at clock during the 1 st 30 minutes of each hour? That would double the chances on you seeing double numbers. Maybe you hear a bell at the beginning of each hour and when you get a chance you look at a clock(??). Are you trying to look at the clock to see double numbers? My brother can tell you the time and be within 10 minutes. So if he tried to look at a clock when he thinks it will be double numbers he will get it correct 1 in 10 times (if he tries once per hour).
 
12:12
01: 01
02: 02
03:03
04:04
05:05
06:06
07:07
08:08
09:09
10:10
11:11

There are 24 possible (AM, PM),ways the hours and minutes have the same digits.

How many total hour:minute combinations are there? [imath]24 \times 60 = 1440[/imath]

So the probability that you'll see [imath]xy:xy[/imath], [imath]P(xy:xy) = \frac{24}{1440} = \frac{1}{60}[/imath].

A frequency of [imath]\frac{3}{10}[/imath] = [imath]\frac{18}{60}[/imath]

You're seeing "double-digits" [imath]18[/imath] times more frequently than would be expected if this was random chance.

However, notice that these "double-digits" are limited to the first [imath]12[/imath] minutes of every hour.

[imath]\frac{3}{10} = \frac{39}{130}[/imath]

For every hour, there are 13 possible minutes less than the 12th minute: 00 to 12. There are 24 hours. Of these only 24 are valid options. So the probability of a "double-digit" = [imath]\frac{24}{24 \times 13} = \frac{1}{13} = \frac{10}{130}[/imath].

Your're only x 3. 9 away from ("better than") chance. Less alarming!

Now, it's [imath]\approx 100 \%[/imath] that this "double-digit" will occur in the morning (9 AMish) and at around lunch time (1 PMish) and so that leaves us with [imath]3 - 2 = 1[/imath], just the [imath]1[/imath] time when it was nothing but luck. Your real frequency (of a "double-digit") then is = [imath]\frac{1}{8} = \frac{13}{94}[/imath].

Compare [imath]\frac{13}{94}[/imath] to [imath]\frac{1}{13} = \frac{8}{94}[/imath], just × 1.625 "better than chance".

If you consider the fact that the 3rd time you saw a "double digit" was at around 5 PM (you leave office), you saw double-digits exactly [imath]3 - 3 = 0[/imath] times (by luck). Compare that to [imath]\frac{1}{13} = 0.07...[/imath]

Good luck!
 
Agent Smith said:
12:12
01: 01
02: 02
03:03
04:04
05:05
06:06
07:07
08:08
09:09
10:10
11:11

There are 24 possible (AM, PM),ways the hours and minutes have the same digits.

How many total hour:minute combinations are there? [imath]24 \times 60 = 1440[/imath]

So the probability that you'll see [imath]xy:xy[/imath], [imath]P(xy:xy) = \frac{24}{1440} = \frac{1}{60}[/imath].

A frequency of [imath]\frac{3}{10}[/imath] = [imath]\frac{18}{60}[/imath]

You're seeing "double-digits" [imath]18[/imath] times more frequently than would be expected if this was random chance.

However, notice that these "double-digits" are limited to the first [imath]12[/imath] minutes of every hour.

[imath]\frac{3}{10} = \frac{39}{130}[/imath]

For every hour, there are 13 possible minutes less than the 12th minute: 00 to 12. There are 24 hours. Of these only 24 are valid options. So the probability of a "double-digit" = [imath]\frac{24}{24 \times 13} = \frac{1}{13} = \frac{10}{130}[/imath].

Your're only x 3. 9 away from ("better than") chance. Less alarming!

Now, it's [imath]\approx 100 \%[/imath] that this "double-digit" will occur in the morning (9 AMish) and at around lunch time (1 PMish) and so that leaves us with [imath]3 - 2 = 1[/imath], just the [imath]1[/imath] time when it was nothing but luck. Your real frequency (of a "double-digit") then is = [imath]\frac{1}{8} = \frac{13}{94}[/imath].

Compare [imath]\frac{13}{94}[/imath] to [imath]\frac{1}{13} = \frac{8}{94}[/imath], just × 1.625 "better than chance".

If you consider the fact that the 3rd time you saw a "double digit" was at around 5 PM (you leave office), you saw double-digits exactly [imath]3 - 3 = 0[/imath] times (by luck). Compare that to [imath]\frac{1}{13} = 0.07...[/imath]

Good luck!
Click to expand...
I must totally disagree with your work. When I'm drunk, I see double-digits all the time!
 
If the probability of seeing double digits = d.

In t trials you should see roughly dt double digits.

If you actually see x double digits, there's a problem only if ... x > dt + [imath]3 \sigma[/imath] or x < dt [imath]3 \sigma[/imath].

The probability of x < 2.5%
 
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