## Amazing coincidence! What are the odds?

This post is by Phil Price, not Andrew

Several days ago I wore my cheapo Belarussian one-hand watch. This watch only has an hour hand, but the hand stretches all the way out to the edge of the watch, like the minute hand of a normal watch. The dial is marked with five-minute hash marks, and it turns out it’s quite easy to read it within two or three minutes even without a minute hand. I glanced at it on the dresser at some point and noticed that the hand had stopped exactly at the 12. Amazing! What are the odds?!

I left my house later that morning — the same morning I noticed the watch had stopped at 12 — to meet a friend for lunch. I was wearing a different watch, one with a chronograph (basically a stopwatch) and I started it as I stepped out the door, curious about how well my estimated travel time would match reality. Unfortunately I forgot to stop the watch when I arrived, indeed forgot all about it until my friend and I were sitting down chatting. I reached down and stopped the chronograph without looking at it. When I finally did look at it, several minutes later, I was astonished — astonished, I tell you! — to see that the second hand had stopped exactly at 12.

I started to write out some musings about the various reasons this sort of thing is not actually surprising, but I’m sure most of us have already thought about this issue many times. So just take this as one more example of why we should expect to see ‘unlikely’ coincidences rather frequently.

(BTW, as you can see in the photo neither watch had stopped exactly at 12. The one-hand watch is about 45 seconds shy of 12, and the chronograph, which measures in 1/5-second intervals, is 1 tick too far).

This post is by Phil.

1. Dustin Fife says:

And to add to the list, I just saw this post at EXACTLY 6:00, which is EXACTLY HALF of 12.

I’m astonished. I wonder what the universe is trying to tell us. Probably something about sports or politics.

2. Yuling Yao says:

Under the null-hypothesis of uniformity, the probability of such 45+1/2 seconds gap is (45+1/2) / 3600*12=0.001, which is 10 times larger in order of magnitude than the p-value of having 45 male presidents in a roll (3e-14 if you need the number).

• Andrew says:

Yuling:

No, you miscalculated the p-value! Don’t forget, Pr(woman)=0.52, so the correct p-value is 0.48^45 = 5e-15. Statistics is hard.

• Ben says:

You unfairly criticize Yuling here. Phil told us the measurement error of his device was 2-3 minutes and then promptly reported “The one-hand watch is about 45 seconds shy of 12”. Bag on Phil. Poor Yuling.

• Andrew says:

Ben:

Remember the research incumbency rule: The first thing that’s published, we assume is true. All critics are held to a higher standard. Indeed, I’m thinking we should criticize Yuling for bullying by daring to criticize a peer-reviewed blog post without first sending his confidential criticisms to Phil on the Q.T. I’m sensing a tone problem here. Who does Yuling thing he is, the replication police?

• Phil says:

Who says Yukon’s didn’t contact me privately? He sent me an email asking for more information about how frequently I wear a watch and what times of day I wind it. I promised to send the data but of course I followed standard procedure by failing to do so.

• Thomas says:

At age>70, or whenever it is that someone becomes a viable presidential candidate, Pr(woman) much exceeds 0.52

3. Jacob Humphries says:

While looking at this I noticed the instantaneous download speed of my phone, when I glance at it, was 129k/s, I. E. It started with “12”. Then I realised not only is it now December, or the 12th month of the year, but it is also the 2nd day of the year (here in Australia). If I list all days of the month of the current month up to today, that gives me ‘1’, ‘2’, I. E. ’12’. Everywhere I look I now see ’12’. What are the odds? What’s the p-value for that? PS. These are not serious questions. Cheers! :-)

4. Rahul says:

Could there be a mechanical explaination for this? Is there something about the internal clockwork that makes the 12 o clock position a “low energy state”?

• Could be but it wouldn’t explain why Phil stopped the stopwatch at 12…

• Phil says:

If the watch had a date wheel, or even if the movement (the internal workings) had the machinery to support a date wheel, then there is some point on the watch dial at which the extra machinery is engaged, leading to higher friction. For most watches it takes some time for that change to happen — the date doesn’t instantly flip from one date to the next. The watch can be set up so the date change starts at any time; sometimes it’s set up to start at 11:45 pm, sometimes right at midnight, etc. If the watch is set up to start the date change exactly at midnight then yes, there is some extra friction starting right then, making the watch more likely to stop then if it was about to stop anyway. And for the entire period that the date is changing, the frictional forces are larger, so the watch would be more likely to stop in that interval (in the sense that more of the mainspring’s energy is used during that interval than any other interval of similar duration, so if the watch is wound at a randomly selected time it is more likely to stop in that interval than another). This doesn’t come into play in this instance — the movement in the Luch doesn’t support a date wheel or any kind of date change — but it’s an effect for most watches. Not a big effect.

For the chronograph there is no such effect, it stops when I push the pusher.

But of course, the watch doesn’t actually get wound at a random time, either. I wind the watch fully when I put it on (usually in the morning, but not always), and then sometimes during the day I’ll idly twist the crown once or twice or a few times.

I don’t know the duration of the ‘power reserve’ on the Luch, but there are several mechanical watch movements with a power reserve of about 40 hours, meaning the watch will run for about 40 hours after being wound completely. Suppose you put on such a watch at 8 a.m. What’s 40 hours after 8 a.m.? Midnight. If you have a watch with a 40-hour power reserve and you typically wind it and put it on around 8 a.m., having it stop within 40 seconds of midnight is really not surprising at all.

5. Dan F. says:

This is the birthday problem. The probability that somewhere someone’s two watches stop at the same second is much greater than the probability that this happens for Phil’s watches. The surprise comes from the selection – Phil observed that it happened to his watches – of course someone wins the lottery, but the probability it is Phil is low.

6. Thomas says:

The 2nd dial has 5 hands, and only one stopped on 12. “12-hacking” at work…

7. AER says:

My rough guess (as someone who hasn’t had a math class in 15+ years) about the probability of something similar happening:

The first watch has a maximum of about 43,200 possible positions (12 hrs * 60 min * 60 s). If it stopped at 45s off, then at least 90 of those positions are relevant (stopping at 12 +/- 45s). Assuming it’s equally likely that it stops at any position (obviously it’s not, in reality, since it depends on when it was wound), then the likelihood of it stopping where it did, or equally close or closer, is 90/43,200 = 0.002083.

For the second watch, it sounds like there are 300 possible positions for the second hand (5 per second x 60 seconds), of which 3 are relevant (stopping at 12 +/- one tick). For the other two hands, it looks like there are 60 possible positions, of which only one is relevant (stopping at 12). So the probability of any one or more hands stopping at a relevant position is (3/300) + (1/60) + (1/60) = 0.043. So the total probability of at least one hand of each watch stopping where they did or closer, assuming they are equally likely to stop at any time, assuming each hand is equally likely to stop at any time, is 1/(0.002083 * 0.043333) = one in about 11,077.

Ok smart folks, please tell me all the ways that I’m wrong :-)

• This is a fine calculation for a particular random number generator, ie. the “null hypothesis” of “equally likely to stop at any time”. Why are we checking that RNG though? I mean, is it really relevant to the question? as you say “obviously it’s not, in reality, since it depends on when it was wound”.

This nicely illustrates the difference between how we do stats when NHST is involved, vs when Bayes is involved. We can “reject the hypothesis” that the watch stopped randomly and uniformly according to your model. Note however, that we can reject that hypothesis no matter where it stopped because it only depends on the width of the interval you’re looking in ;-)

Of course, in reality, that’s not how watches work. First off, the one that ran out of spring tension and stopped will run from the last time it was wound for a number of seconds that depend on things like the amount of energy in the spring and the friction in the movement. For that watch, put a prior over when the watch was wound, and the number of seconds that it can run before stopping when wound. Collect some data on when Phil winds his watch, and how long it runs when wound (a gamma distribution is probably appropriate for a likelihood), then calculate a posterior probability to be within epsilon of the 12 position.

For the second watch, it was stopped by Phil after his travel, so we need a prior on the time it takes him to make the trip (perhaps again a gamma distribution), plus the amount of time it takes him to remember to stop the watch after his trip (another gamma distribution?). Then collect some data and calculate a posterior probability.

The relevance of these calculations is obvious because there’s a clear connection between the design of the generating process and what actually happens in the world.

One suspects that these probabilities are more concentrated around what happened than your uniform examples, so you could consider your uniform examples as lower bounds on the probability. My guess is the generative model calculations would be easily a factor of two larger.

8. Kindred says:

You’re pretty deep into the watch nerdery if you have both a Luch and a Seagull 1963.

Those Seagull movements are pretty great. State of the art Swiss factory watchmaking, c. 1955. But still cool to look at, and fairly reliable all things considered.

• Phil says:

I own a bunch of watches but I don’t think of myself as a watch collector, just as I own a bunch of shirts but don’t think of myself as a shirt collector. I have a fondness for slightly quirky things, and an appreciation for elegant mechanical solutions to problems, and I like being able to check the time without pulling my phone out of my pocket, and those all intersect to make me a bit of a watch nerd, although nowhere near what a lot of watch collectors are like.

The Luch was about \$30 and I love it.

Another quirky watch is my Russian 24-hour watch, which has a rotating bezel to keep track of a second time zone. I bought this to go with the 24-hour clock in my office, which has a polar projection map that rotates with the hour hand so you can see the time anywhere in the world at a glance. Back before I had a cell phone and companion smart watch I used to travel with this 24-hour watch and use it as intended, not because I really needed to but just because that sort of thing is my idea of fun.

The reissue of the 1963 Chinese chronograph cost a lot more, maybe \$300 or something. I got one for my brother for his birthday (he was born in 1963) and liked it so much I got one for myself too. Very distinctive look. And yes, the movement (visible through the display case back) is beautiful…and just such a fine bit of engineering, a column wheel chronograph. I love the fact that you can see how it all works.

Kindred, since you chose that moniker I’m guessing you won’t be too bored if I tell you that I also have two, count em two, Bulova men’s dress watches, bit in elegant gold-plated cases. They’re men’s watches but are tiny by today’’s standards, even for women’s watches. I got the first on eBay for \$23, with the idea of wearing it to special hoity-toity occasions like the opera. It was made in 1946 from prewar parts. For my purpose I only need it to keep time within about a minute per hour. — put it on at 5 pm, take it off at 11 — but it was even worse than that so I bought a similar one (from 1955 or so) for about \$50 that works fine. But I couldn’t bring myself to toss the old one, and ended up having it cleaned, which cost twice as much as buying the two watches in the first place. So now I two tiny, elegant dress watches.

I also have a quartz Seiko that is close to my platonic idea of watch design, and I’ve seriously considered (but rejected) spending thousands of dollars to buy a mechanical watch with similar looks and functionality. It just seems..I dunno, inelegant…to have to buy a battery every few years for a watch I only wear a few days per month.

To round out my ‘collection’ I also have a bronze blue-dial dive watch from Zelos. This one is my favorite daily wear watch.

9. The bigger coincidence is that Andrew also wears a one-handed watch.

• Well, Phil and Andrew have been friends since high school right? So what’s the chance that Phil gave Andrew the watch? my prior is it’s a nontrivial probability.

• Phil says:

Andrew and I are both fond of unusual timekeepers — my 24-hour clock with a map that rotates, Andrew’s clock that runs backwards — so I described my one-hand watch to Andrew and showed him a photo of it. He drooled with envy and ordered one for himself. He said his doesn’t even run for a full day on a wind, though.

Anyway, not much of a coincidence.

And yes we’ve been friends since high school, but we have also been friends since junior high school!