I'm here at the Guilford Parkrun. Behind me there are a lot of people
about to run precisely five kilometres.
It's not just happening here.
There are hundreds of Parkruns across the UK, thousands internationally.
And because it's such a precise distance that lots of people are running and
you can track your stats across every single park run you go to,
it attracts a lot of stats nerds.
Speaking of stats nerds, I'm here with my friend Geoff Marshall,
who you may know from the YouTube channel
Geoff Marshall.
And Geoff sent me an email, possibly a year or two ago, pre-pandemic,
and said, "there's a thing called Parkrun Bingo."
When you get your official finish time, not what your watch says,
but you get a position and a time,
And if you take the seconds part of your time
(so, say you finished in 24 minutes and 12 seconds, you take that 12)
There's like a game, an unofficial challenge.
Can you tick off to say you've done all 60 positions, from 0 through to 59?
And have you done this? It took me, we worked it out,
We were on a train somewhere, on the day we did platform zero.
It took me 229 runs to tick off all 60, but I gather on average-
Well, how what? No spoilers, no spoilers.
Geoff did it in 229.
The question is, is that above or below average? We're gonna run the numbers on this.
But the thing is, you've already done it,
Which means there's no narrative for my video. GEOFF: So I've brought some friends.
You brought some friends who have not done it.
And I've got- I've printed out what times they have and have not yet achieved.
So you and your mates are gonna run the Parkrun? Yes.
I'm gonna do the maths of if you're above or below average,
and then we'll see, this is the jeopardy,
if any of your friends get a new time.
I'm here with friends of Geoff
Emily, you're only missing, like five.
That's amazing. So if you could aim for, like, the 12 and a half
I think that's the plan and maybe get the 16 instead.
Yeah, exactly. You can overshoot slightly. You've got strategy.
I've also printed out all the numbers for Matt.
And Matt, you've got a running channel. I do yes. So Runderground.
Highly recommended. You're missing, like 10.
Something like that. Do you know what, I've got it all prepared.
I've got it all written down. You've got them in your hand!
This is how serious I'm taking it!
So good. Okay, alright,
Mark, you're exactly halfway. You've got 30 of 60.
There's a 50:50 chance you'll get a new one by the end.
I'll let these guys go and prep and I want to go to some calculations.
Parkrun bingo is actually an example of something called stopwatch bingo,
where you're just trying to collect all the different seconds,
assuming this happens at random.
And it's an example of what's called the coupon collector's problem.
That's where you've got n things you want to collect, and you want to get all of them.
However, not only do you pick them at random, but you've got replacement,
so you could always pick the same one again later on.
The question is, how many times, on average,
if you want to collect n coupons or in this case, n seconds in your times,
how long is that gonna take, like,
how many times are you going to have to do that before it happens?
Of course, we can work it out
with mathematics. Before we do that,
just a quick sense-check.
For the very first time you do a parkrun, you're going to get a new time.
And the second time you do it,
you're probably gonna get a new time,
because you've still got 59 of the 60 not yet collected.
And as you go further and further into it,
it becomes gradually less and less likely until-
Until finally, at the very end,
you've only got one time outstanding.
You've got 59 of the 60, and you're thinking, well, surely,
I'm going to complete my parkrun bingo any moment now.
The issue is there are 60 seconds in a minute,
so there's 60 different values you have to collect.
However, there are only 52 weeks in the year,
and 60 is a bigger number than 52.
So you think I've only got one left, but on average,
it's gonna take you over a year of doing
weekly Park Runs before you get that very last value. I'm aware I've put
a lot of props on the table I probably didn't need
to make the very simple point that 60 is bigger than 52.
In fact, what I've done is, on my table, I've got here
I've put unnecessary props. Allow me to demonstrate.
So I've put the calendar on this side of my table.
And then I put my clock on this side of the table,
and all these props weren't strictly needed to make an
otherwise very simple point, that I've got too many props.
What I'm trying to say is some people run parkruns and
some people make unnecessary scale models for a recursion joke.
And they're all perfectly valid hobbies.
Ah, you know I did. There it is. Okay, so there's the next one down another 10% scale.
I think that was definitely worth the extra effort.
Quick sense-check before we start.
If we come up with a formula for n coupons to give us the expected number of trials,
that should be a formula which keeps getting bigger and bigger
and bigger because as you have more and more coupons,
it's always gonna take a bit longer to get them all.
So the formula we come up with shouldn't approach- it shouldn't converge on a limit.
It should be some kind of formula that keeps getting bigger forever.
And a thing to bear in mind is, if something has a probability of p of happening,
then the amount of time you normally have to wait for that to occur,
so the expected time you wait, equals one over p.
It's the inverse of that probability.
So, for example,
if there's a 1 in 52 chance of picking a card at random out of a deck
and then you start doing it over and over again,
to try and get a specific card,
then that will take you, on average 52 times.
Now, this may sound obvious, but to get all the coupons,
you have to get all the coupons.
So for coupons, "all" that means you've got to get the first coupon
and then you've got to get the second coupon.
Then you've got to get all the way up to the i-th
coupon and then you carry on all the way until the n-th coupon,
and then you've got all n of them and by coupons.
I'm just being the times people are getting here. So actually,
the total time required is the time to get the first coupon.
Then you add on the extra time it's going to take to get the second one.
And then you add on the extra time it's gonna take to get the next one
up to the i-th one all the way up to the n-th one.
So what we have to do is work out.
How long will it take us to get the i-th one,
and then sum that from the first one all the way up to the n-th one?
One quick clarification when I'm saying the first coupon, the second coupon,
etc, I don't mean the values of the coupons or, in this case,
the value of the seconds by first coupon
I just mean the first one you happen to collect. It could
be any of the coupons, it's the first one to you.
Okay,
as you were. Okay, Quick example.
Just to get a sense of the i-th coupon, let's say we do
Emily's. Emily has done 55 so she's currently after i equals 56 the 56th coupon.
Now,
to get the 56th coupon,
there's only
five left that she hasn't got,
so the probability is going to be the five left that she hasn't got any of those out of
the 60 total ones.
Well, how do we get five?
Well we knew
that the 56th coupon means there's 55 she's already got and then this value up here,
what we did was we took 60
the total number, subtract how many she already had. So we subtracted 55.
So if you want to do that in
general than what we've done is the probability of getting coupon i
equals the total number you could have,
so n, subtract how many you've already got and you've got one fewer than the i-th value.
So that is, i subtract one and it's out of n.
Oh, and I could simplify that.
I could take the brackets out that would be n minus i plus 1.
Actually know what let's pop that down here.
So the probability is actually n minus i plus 1 all over n
which means the time it takes to get the i-th coupon is just the inverse of that.
So it's n divided by n minus i plus 1 and we are almost there,
just like almost everyone has finished their park run.
I'll get to the conclusion now.
So
the total time
is going to equal
the first coupon,
plus the second one,
plus the next one all the way down.
And instead of doing this back in our general term,
I'm now going to do it specifically for a Park Run.
So the time for the very first one is the inverse of the first probability,
which is 60 out of 60.
Well, that's easy. 60 out of 60
Then the next one is the inverse of 59 out of 60. 60 over 59.
And then the next one is the inverse 60 over 58.
And then that carries all the way down until when you're finally finishing
you've got 60 out of 3, plus 60 out of 2,
plus 60 out of 1. Which I can rearrange to put the 60,
that's the n, out the front. And, oh!
you know what?
I'm just going to reverse the order of these.
So if you flip them around and I'm going to start by writing this, the last one first,
then it's 1/1 plus a half, plus
a third plus all the way up to
1/60. So
it's 60
outside of the harmonic sequence. It's 60 at the front of every single unit
fraction being summed.
So actually, we want to put that back into general terms. It equals n outside of
the harmonic series,
all the way up to the n-th term in the harmonic series.
What we said at the beginning is we expect whatever this
formula is for the number of attempts it's going to take
to always increase forever and n is- n is finite.
And so we've just managed to prove that the harmonic series diverges.
It's proof by running around a park.
So if you actually run the numbers on all of that for
a Park Run for the harmonic series up to 60 the total time
equals- I don't want to judge, but I feel like we're reaching
the end of the field of runners.
Uh, equals 181 rounded to the nearest whole number of park runs.
oh sorry, 281
round to the nearest whole number of Park Runs,
and Geoff did it in fewer than that. So there you are. Geoff is below average.
Alright, let's go close off the narrative, see what we got.
Yeah. Uh, Matt,
How many seconds? First of all, how was your time?
Okay, so I'm happy with the time. I had a one in 10 chance
and I thought it was getting towards like the 21 thing.
MATT: Yeah So it's all written down there
So, 21:43. I missed that. 21:47, missed that.
21:49. I missed that, but the finish was in sight.
I just got over the finish was 22:02.
My watch
said 22:03
MATT: The official time? GEOFF: But what happens... Waiting for the officials.
We've handed in... You hand in your token and your scanner. They now go away.
MATT: So we don't know yet? So Matt you've got to patiently wait.
A couple of hours for the run director- MATT: A couple of hours?
What happens at about midday.
You get sent... a text on your phone appears and everybody jumps on the website.
and that's when the official times come through.
So quite often your time- MATT: That's not, that's not how YouTube works.
We need closure now-
So join us after the break when Matt
will be looking slightly less disappointed MATT: Sometime later
as well as the official time.
Oh, my goodness.
All right. Geoff has finished his run,
and I've actually got; this is the graph I made for you on that train trip way back,
back, back in the day.
And you can see at the beginning here, you start off really nice and linear.
Bam, bam, bam, bam, bam! One after the other.
Took me about until 17 to get my first repetition. Yeah.
Then you've got a few reps in a row and then this is still not far off linear
I had a little nice run there. I had about 10 with no repetitions.
And then another drought and gradually flattens out, flattens out, flattens out.
And actually, at the end here, you did- uh, it's only
Like 29 or 30? GEOFF: I think it was about 229 I think.
No. but on your last one, to get your last one,
you did it in way faster than you would expect.
What are the chances of the last one
as a percentage?
Well, it should take you on average another 60 runs to get the very last one.
And you did it in, I think under 30.
So you're you're way ahead. And overall,
the expected
number of runs for the whole thing is 281.
So I was 229. I was ahead way ahead, way ahead, way ahead.
But some poor people do like 500 runs and never, never quite make it. MATT: I am curious
because you're telling me this attempt, like when people are trying to
game the system and and and like, they'll slow down and speed up, it's called sandbagging
and it seems quite prevalent.
Not gonna lie.
I feel like a lot of- and were you sandbagging
in some of these? GEOFF: I may have. I remember needing 57 seconds
and part of me's like "If I slow down a little bit..."
MATT: Here's probably a clump of your times around 57. GEOFF: Very last one.
It might be a little clump around the 57 mark
on the very last one.
I think it would be interesting to look
at the actual stats and see because the theoretical
number of attempts is 281.
In practise,
I suspect it's going to be- GEOFF: so take out the time when I did it and see how long it takes
to actually do it. MATT: I think if you got
a lot of people's data,
you'd get a bi modal distribution because there'll be one peak early on
for people who have been sandbagging and trying to get the full bingo.
And there'll be another peak for people who are
blissfully unaware of the notion of this bingo.
And there they'll be at 281-ish and there'll be another
more spread out one earlier on a
GEOFF: They're like the pure control, yeah
I'll plot those stats afterwards, if I can find them. GEOFF: So next time you
come and do the run?
Nah, I've hung up my- hung up my uh
I remember when I was a runner.
My knees didn't take it, so I switched to cycling. GEOFF: Cycling?
GEOFF: Much more healthy. MATT: Low impact.
GEOFF: Thanks for coming to Guilford. Thanks for... MATT: My pleasure.
GEOFF: Thanks for taking 20 months to reply to my email. MATT: Geoff Marshall.
He's got a YouTube channel.
Thanks for watching the video.
If you like transport infrastructure videos,
you absolutely should check out Geoff's Channel.
And if you enjoy socially running for precisely 5,000 metres,
do check out your local Park Run.
But now
I feel like we need some closure.