# Thread: The probability of seeing 13 minutes past the hour at least once every day?

1. ## The probability of seeing 13 minutes past the hour at least once every day?

Each day, there are 24 opportunities in a 24 hour period to see 13 minutes past the hour.

There are 60 x 24 possible times that can be reported in a 24 hour period (1440 combinations).

Probability of seeing 24 of these combinations:

24
____ = .016 or 1.66% of the time you would see 13 minutes past the hour.

1440

But shouldn't the odds improve when you only stay awake for only 16 hours?

There are 16 opportunities in a 16 hour time period to see 13 minutes past the hour.

There are 60 x 16 possible times that can be reported ( 960 combinations).

Probability of seeing 16 of these combinations:

16
____ = .016 or 1.66%

960

Can anybody point me in the direction of what I'm doing wrong?

2. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

The numerator is a proportion (directly related to) the denominator. So if you change the denominator you change the numerator as well, as long as you are talking about whole hours your % will always be the same.

Perhaps simplify the fractions down and it will make sense:

24/1440 =1/60

16/960 = 1/60, etc...

3. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Thanks for a response, hlsmith.

I see the numerator decreases with the denominator in such a way that the probability is always the same even down to the probability of seeing 13 minutes past the hour in 1 hour:

1/60 = .016

What about the method itself? Is it a valid way to determine the probability of seeing 13 minutes (or any specific minutes) past the hour ?

4. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

You are talking about a event that exists 1/60 of the time over
extended time periods (hours). The probability that you will witness
that event when you happen to glance at a clock is 1/60
regardless of how many hours you sleep (within reason),
providing, of course, that your glances occur purely by
chance (at random).

Art

5. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

As note before this is also given that the denominator is always 60 minutes. If your time period changes above or below 60 minutes after simplifying the fraction, the probability will change.

6. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Originally Posted by hlsmith
As note before this is also given that the denominator is always 60 minutes. If your time period changes above or below 60 minutes after simplifying the fraction, the probability will change.
When would there be a case that the denominator doesn't reflect all the minutes in an hour (60) ? The question is: What's the probability of seeing 13 minutes past the hour at least once every day?

One other question (related, not the same as above): Consider a conditional probability, such as "What's the probability of seeing 13 minutes past the hour every day GIVEN that you look at the clock 12 times a day?

Isn't the answer to this:

1. The probability of each individual chance you have to see 13 minutes past the hour = .016 (from above)

2. There are 12 chances, so: .016 x 12 = .192

3. The probability of seeing 13 minutes past the hour has increased to 19% of the time given you look at the clock 12 times a day.

Is that right?

7. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

On any given 24 hour day, if you randomly look up at a clock you have a 0.0167 chance of it being 13 minutes after the hour.

You need to make sure you have enough detail in these questions.

On any given 24 hour day, if you randomly look up at a clock 12 different times during 12 different minutes you would have a 0.016 + 0.016,...,+0.016 = 0.19 chance of it being 13 minutes after the hours.

8. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

One more. Related, but a different question:

A person looks at a clock 12 times every day. Each day the person manages to see one time that it's 13 minutes past the hour. What's the probability of this happening each day for a week?

The probility for each day is a mutually exclusive event that doesn't depend on what happened the prior day. But we're talking about consecutive events, which makes the probability go way down, yes?

So I think you have to do this:

a. Probability of seeing 13 minutes past the hour GIVEN you look at the clock 12 times per day = .19 (from above)

b.The question involves a consecutive event for 7 days, so now the probability of seeing 13 minutes past the hour EVERY day for a week requires you to multiply the probability for each day 7 times:

.19 x .19 x .19 x .19 x .19 x .19 x .19 = .000008

= .8 x 10^-5 or .0008%

Is that right?

9. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Just to make sure that you understand what hlsmith emphasize:

if you randomly look up at a clock 12 different times during 12 different minutes
The point is like you randomly choose 12 different (distinct) minutes (hour is irrelavant here) and you will have the sum of the probabilities as they are mutually exclusive events.

The situation is completely different if you want to choose 12 moments to check the clock independently. In such case the probability will be

10. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Originally Posted by BGM
Just to make sure that you understand what hlsmith emphasize:

The point is like you randomly choose 12 different (distinct) minutes (hour is irrelavant here) and you will have the sum of the probabilities as they are mutually exclusive events.
--Mutually exclusive for an hour and for the day, yes. This is the case where throughout the day a person randomly looks at the clock 12 times, each time to see what time it is, which wouldn't need to be twice within the same minute. So you find the probability of seeing 13 minutes past any hour in a 24 hour cycle and multiply by 12.

Originally Posted by BGM
The situation is completely different if you want to choose 12 moments to check the clock independently. In such case the probability will be

--and the above takes into account that some of the 12 moments will be within the same minute, yes?

What about the case I illustrate above (post #8) where the question is: The probability of seeing 13 minutes past the hour every day for a week? Is that wrong if it's assumed the person never looks at the clock twice in the same minute?

11. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Originally Posted by BGM
Just to make sure that you understand what hlsmith emphasize:

The point is like you randomly choose 12 different (distinct) minutes (hour is irrelavant here) and you will have the sum of the probabilities as they are mutually exclusive events.

The situation is completely different if you want to choose 12 moments to check the clock independently. In such case the probability will be

Very interesting comparison. It's another example of thinking in terms of the p of not happening and then
subtracting that result from one to find the p of it happening. Instead of the fixed exponent of 12 I
played with using N instead. Running N upwards, at N = 480 the p of the event happening rose to
0.99968. At N = 41 there's a 50/50 chance of the event happening. So 41 random glances at the
clock at any time makes p a coin toss

I have to thank BGM for this even though it's not my thread.

Art

12. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Originally Posted by ArtK
Running N upwards, at N = 480 the p of the event happening rose to
0.99968. At N = 41 there's a 50/50 chance of the event happening. So 41 random glances at the
clock at any time makes p a coin toss Art

I would think it's obvious the more times you look at the clock the more likely it is you'll see 13 minutes past the hour, yes?

But when we're talking about seeing 13 minutes past the hour EVERY DAY, whenever you look at the clock (given it's done 12 times), for a period of a week, that's got to be a very low probability. It's going to get needlessly more confusing if the question should be what's the probability of NOT seeing 13 minutes past the hour. Likewise, it's unnecessary to consider looking at the clock in moments vs. minutes. In the interest of knowing what time it is, nobody needs to look at a clock twice in the same minute. I'm talking about the common circumstance of someone looking at the clock throughout the day.

In case it's not clear (because people are telling me I'm not providing enough detail), I'm trying to find three different but related probabilities:

1. Probability of seeing 13 minutes past the hour in a day. (post #1)

2. Probability of seeing 13 minutes past the hour in a day given that you look at the clock 12 times a day. (post #6)

3. Probability of seeing 13 minutes past the hour every day for a week given that you look at the clock 12 times a day. (post #8)

I 'm confident about #1, I see I was wrong about understanding the alternate method for #2, and I'm unsure about #3, though I think I attempted a solution.

13. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

it's unnecessary to consider looking at the clock in moments vs. minutes.
nobody needs to look at a clock twice in the same minute.
Actually these related several key questions in this problem:

1. Everyone in this thread presume from the very beginning that people are randomly looking the clock uniformly. If that is not the case the calculations in all above posts will be wrong.

2. How long do we look at the clock? We all assume that it is done instantaneously, a 0 second length time interval. Again the argument will be wrong if we look at the clock for a period of time and consider the event happen if the time interval have any portion of it overlapping with the 13th minute. Maybe we are ignoring this part if you have already thought that it is a discrete problem; But to me a random time is like a continuous random variable.

3. Most importantly, you need to know whether multiple random looking time are mutually independent to each other. It will be different if they are mutually exclusive. For independent case, we will have to independently choose, say 12 random times within a given time period before we know that what time we have chosen. So it is entirely possible we have chosen the same minute more than once.

14. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Originally Posted by BGM
Actually these related several key questions in this problem:

1. Everyone in this thread presume from the very beginning that people are randomly looking the clock uniformly. If that is not the case the calculations in all above posts will be wrong.

2. How long do we look at the clock? We all assume that it is done instantaneously, a 0 second length time interval. Again the argument will be wrong if we look at the clock for a period of time and consider the event happen if the time interval have any portion of it overlapping with the 13th minute. Maybe we are ignoring this part if you have already thought that it is a discrete problem; But to me a random time is like a continuous random variable.

3. Most importantly, you need to know whether multiple random looking time are mutually independent to each other. It will be different if they are mutually exclusive. For independent case, we will have to independently choose, say 12 random times within a given time period before we know that what time we have chosen. So it is entirely possible we have chosen the same minute more than once.
Sure. It's also more than likely that we won't get a "hit" for many days. Using p = 1 - (59/60) ^12 = 0.182 all we know is that over thousands of days we'll get hits on 18% of the days, assuming the random sampling is spread out over the course of a day (no more often than 1 sample per hour). Concerning #2, I think a "glance" at a clock takes less than half a second so the error due to this finite sampling time could be in the 1% range.

Art

15. ## Re: The probability of seeing 13 minutes past the hour at least once every day?

Originally Posted by BGM
Actually these related several key questions in this problem:

1. Everyone in this thread presume from the very beginning that people are randomly looking the clock uniformly. If that is not the case the calculations in all above posts will be wrong.
"People" are not looking at the clock. It's a probability that pertains to one person.

If I really need to be more clear:

1. Probability of a person seeing 13 minutes past the hour in a day, assuming only one look at any given minute (i.e., a mutually exclusive event between minutes). (post #1)

2. Probability of a person seeing 13 minutes past the hour (at least one time) in a day given that the person looks at the clock 12 times a day, each look is a mutually exclusive event. (post #6)

3. Probability of a person seeing 13 minutes past the hour (at least one time) every day for a week given that the person looks at the clock 12 times a day, each look is a mutually exclusive event. (post #8)

Originally Posted by BGM
2. How long do we look at the clock? We all assume that it is done instantaneously, a 0 second length time interval. Again the argument will be wrong if we look at the clock for a period of time and consider the event happen if the time interval have any portion of it overlapping with the 13th minute. Maybe we are ignoring this part if you have already thought that it is a discrete problem; But to me a random time is like a continuous random variable.
The above is noted, but beyond a miniscule detail of concern. The probabilities are defined as a common look at the clock, not a continuous stare lasting seconds.

Originally Posted by BGM
3. Most importantly, you need to know whether multiple random looking time are mutually independent to each other. It will be different if they are mutually exclusive. For independent case, we will have to independently choose, say 12 random times within a given time period before we know that what time we have chosen. So it is entirely possible we have chosen the same minute more than once.
I already said above that each event of looking at the clock is a mutually exclusive event. The randomly choosing that you refer to (i.e., includes times when you will look twice in the same minute) does NOT comply with how a person commonly looks at the clock throughout the day to see what time it is. Again, it's a needlessly convoluted take on what I'm asking.

I would think being wrong by maybe 1% for a probability that (I still don't have any confirmation on this, post #8) is only about .0008 is acceptable, yes?

But tell me please if I'm still wrong about the method above for determining the probability of ONE PERSON seeing 13 minutes past the hour every day for a week given the person looks at the clock 12 times per day, with each glance at the clock assumed to be a mutually exclusive event.