Hypothetical Flat-Tire Question

On 17 Jan 2004 18:55:42 -0800, [email hidden] (Carl Fogel) may

have said:

How many of these stickers should I sprinkle randomly to give careless Carl a flat tire? And how
many more should I sprinkle to deflate Hawkeye's alert ego?

Okay, then for any given point on the track where there is a single goathead, assuming mere
random placement of both the tire and the goathead, the chance of each tire intersecting the
goathead is 1 in
100. The chance of a bike hitting a specific goathead is roughly double that, because it is given
that the rider weaves around a lot, which means that the two tires follow essentially
independent paths. I will treat the problem as being 200 events of 1% probability with the
rider's subsequent record irrelevant after a hit.

This does not mean that with 100 goatheads scattered randomly over the length of the course, a 1%
probability per tire, and two tires, that the probability of a hit is 200%. That is not the case.
The non-alert rider's probability with 100 goatheads randomly scattered over that long trail is
86.6% of having hit one.

Regardless of the number of goatheads sprinkled up to a level *less than* total path saturation,
there is a measurable probability of either or both riders successfully avoiding all of them.

OTOH, if 102 are laid out cheek-by-jowl across the path in a .98" spacing line, the lead cyclist
can't fail to hit one, although that's probably pushing it on real-world guarantee of a puncture. No
provision was stated for the alert rider to be able to stop or portage past a goathead, but if
that's a possibility, then there is no absolute-certainty puncture scenario with either random
goathead placement or a solid line.

OTOOH, if neither cyclist is carrying a spare tube and/or patch kit, the probability is high that a
tire will have been flatted before the first goathead can be positioned. (This is due to the
operation of the Laws Of Perversity Of The Universe.)

Quoted message said:

Hint: does the total number of stickers required have anything to do with whether the distance is
one mile, a century ride, or to the moon and back?

It is dependent only upon the total number of goatheads, and at high densities (i.e. more than 1% of
the trail width produces a puncture) their density as a percentage of the width of the trail.

You didn't state whether the initial conditions include the knowledge of the possible presence of
goatheads on the part of the careful rider, and you didn't state whether the goatheads had a
specific probability of not being recognized if either rider was looking straight at them, or of
being hidden by leaves, etc.

Quoted message said:

Extra-credit: given ten thousand stickers, how many stickers do both riders avoid through sheer
blind luck? That is, how much dodging is actually useful?

I could haul out the stats book and come up with several justifiable answers to this, but my
instincts tell me that they won't get six feet down the trail if either of them borrowed my bike for
this ride.

Quoted message said:

Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
question.

Getting loaded before answering may improve the question. It's Saturday, after all, at least in part
of the world.

--
My email address is antispammed; pull WEEDS if replying via e-mail.
Yes, I have a killfile. If I don't respond to something,
it's also possible that I'm busy.
Words processed in a facility that contains nuts.

Carl Fogel said:

Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live where
goathead stickers are unknown.

Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.

Two bicyclists will pedal this path at 20 mph for 5 hours.

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

--
Perre

You have to be smarter than a robot to reply.

On 17 Jan 2004 18:55:42 -0800, [email hidden] (Carl Fogel)
wrote:
Carl's Conundrum:

Quoted message said:

entertain snowbound riders lucky enough to Let's say that the front and rear tires of each bicycle
will sweep a track an inch wide.

Quoted message said:

The other bicyclist--let's call him Hawkeye--is an alert rider with excellent vision who also
weaves randomly along the same path, but enjoys reflexes good enough to swerve around any small
hazard that he sees in time.

Quoted message said:

How many of these stickers should I sprinkle randomly to give careless Carl a flat tire? And how
many more should I sprinkle to deflate Hawkeye's alert ego?

I haven't got the knowledge to tell you how many you need to sprinkle, but I can tell you that it's
the same for Carl as it is for Hawkeye. They will be invisible under the snow; and if they're on top
of the snow, these are the possibilities:
- They'll sink in enough not to be seen
- between color and size, they'll be invisible on top of the snow
- they won't affect either rider because they will penetrate the snow easier than the tire

Of course, none of that matters, as our Skippy eating friend Perre points out, since you've already
told us that they go 100 miles.

Quoted message said:

[email hidden]

--
Rick Onanian

Per Elmsäter said:

Carl Fogel said:

Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live where
goathead stickers are unknown.

Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.

Two bicyclists will pedal this path at 20 mph for 5 hours.

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

Oh. I see how YOU are!

Tim Lines said:

Per Elmsäter said:

Carl Fogel said:

Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
where goathead stickers are unknown.

Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.

Two bicyclists will pedal this path at 20 mph for 5 hours.

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

Oh. I see how YOU are!

huh?
--
Perre

You have to be smarter than a robot to reply.

Per Elmsäter said:

Tim Lines said:

Per Elmsäter said:

Carl Fogel wrote:

>Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
>where goathead stickers are unknown.
>
>Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>
>Two bicyclists will pedal this path at 20 mph for 5 hours.
>

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

Oh. I see how YOU are!

huh?

You're the kind who sees the right answer without knowing WHY it's the right answer. Of course they
got no flats, but WHY?

The answer? Since Carl and Hawkeye are regular readers of rbt, they know that the only proper and
acceptable way of dealing with road debris of any kind is to stop, pick the bike up, and carry it
past the the dangerous area. So that's what they did. They RAN the final 99.9 miles with their bikes
slung across there backs, thereby completing their century while protecting their precious tires
from the onslaught of potentially thousands of sharp thingamajigs.

Obvious, isn't it?

[email hidden] (Carl Fogel) wrote in message news:<[email hidden]>...

Quoted message said:

Quoted message said:

Here's a picture of a real goathead:

home.comcast.netGoathead.jpgOpen ↗

It's sitting on a dime about 0.705" wide.

What kind of a person says "about" 0.705" wide?

I suppose if you describe it to the nearest thousandths of an inch then that's "near enough".

Quoted message said:

Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
question.

[email hidden]

"Per Elmsäter" <[email hidden]> wrote in message news:<[email hidden]>...

Quoted message said:

Carl Fogel said:

Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
where goathead stickers are unknown.

Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.

Two bicyclists will pedal this path at 20 mph for 5 hours.

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

But he doesn't specifcally say that those two bicyclists are Carl and Hawkeye. They could be two
other cyclist riding along the same bicycle path. And if some of the thorns are picked up in the
tyres of the othe bicyclists, does that upset the computations.

Marty

[email hidden] (Marty) wrote in message news:<[email hidden]>...

Quoted message said:

[email hidden] (Carl Fogel) wrote in message
news:<[email hidden]>...

Quoted message said:

Quoted message said:

Here's a picture of a real goathead:

home.comcast.netGoathead.jpgOpen ↗

It's sitting on a dime about 0.705" wide.

What kind of a person says "about" 0.705" wide?

I suppose if you describe it to the nearest thousandths of an inch then that's "near enough".

Quoted message said:

Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
question.

[email hidden]

Dear Marty,

Touché!

Actually, a dime is just a hair under 52/73rds of an inch wide.

Alas, my dial calipers don't have fine enough ends to fit in between the itty-bitty ridges, so I was
forced to use Kentucky windage and honorably added the qualifying "about" lest I be accused of
excessive precision.

(Or was joking.)

So how many stickers do ya reckon, plus or minus half a dozen?

Carl Fogel

Marty said:

"Per Elmsäter" <[email hidden]> wrote in message
news:<[email hidden]>...

Quoted message said:

Carl Fogel said:

Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
where goathead stickers are unknown.

Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.

Two bicyclists will pedal this path at 20 mph for 5 hours.

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

But he doesn't specifcally say that those two bicyclists are Carl and Hawkeye. They could be two
other cyclist riding along the same bicycle path. And if some of the thorns are picked up in the
tyres of the othe bicyclists, does that upset the computations.

Well that would explain how the road was cleared from debris and thorns. Smart thinking.

--
Perre

You have to be smarter than a robot to reply.

Tim Lines said:

Per Elmsäter said:

Tim Lines said:

Per Elmsäter wrote:

> Carl Fogel wrote:
>
>
>> Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
>> where goathead stickers are unknown.
>>
>> Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>>
>> Two bicyclists will pedal this path at 20 mph for 5 hours.
>>
>
>
> Since you just told us they would be pedaling for five hours at 20 mph you have also let on
> that they reached the destination at the end of the 100 mile road without any flats at all.
>

Oh. I see how YOU are!

huh?

You're the kind who sees the right answer without knowing WHY it's the right answer. Of course
they got no flats, but WHY?

The answer? Since Carl and Hawkeye are regular readers of rbt, they know that the only proper and
acceptable way of dealing with road debris of any kind is to stop, pick the bike up, and carry it
past the the dangerous area. So that's what they did. They RAN the final
99.9 miles with their bikes slung across there backs, thereby completing their century while
protecting their precious tires from the onslaught of potentially thousands of sharp
thingamajigs.

Obvious, isn't it?

You should be careful when you start generalizing without having something on your feet. As a matter
of fact you should be very careful even then. Of course I know WHY it's the right answer and if you
learn to read you'll figure out what I said too 😉 Once you've understood what Carl wrote you'll
realze that they didn't run or walk a single step since he clearly states that they *pedaled* for 5
hours going 20 mph.

--
Perre

You have to be smarter than a robot to reply.

Please accept my apology for failing to treat earlier response with the seriousness that it was
apparently intended. My assumption ("I see how YOU are!"😉 was that you take things lightly. A
person's tone is hard to read via USEnet, I clearly missed yours. I should know better than to have
made that assumption. So I'm sorry if you took that as an attack of any kind.

Tim Lines said:

Please accept my apology for failing to treat earlier response with the seriousness that it was
apparently intended. My assumption ("I see how YOU are!"😉 was that you take things lightly. A
person's tone is hard to read via USEnet, I clearly missed yours. I should know better than to
have made that assumption. So I'm sorry if you took that as an attack of any kind.

No problems Tim. I wasn't offended, just surprised at your assumptions. However you are quite
correct in that I take this thread rather lightly as in humorous. Although my solution to the
problem was 100% serious and of course the only correct answer offered so far.

--
Perre

You have to be smarter than a robot to reply.

Per Elmsäter said:

Marty said:

But he doesn't specifcally say that those two bicyclists are Carl and Hawkeye. They could be two
other cyclist riding along the same bicycle path. And if some of the thorns are picked up in the
tyres of the othe bicyclists, does that upset the computations.

Well that would explain how the road was cleared from debris and thorns. Smart thinking.

Obviously the answer is to always ride behind steamrollers in thorny conditions.
--
David Damerell <[email hidden]> flcl?

[email hidden] (Carl Fogel) wrote in message news:<[email hidden]>...

Quoted message said:

[email hidden] (Marty) wrote in message news:<[email hidden]>...

Quoted message said:

[email hidden] (Carl Fogel) wrote in message
news:<[email hidden]>...

Quoted message said:

Quoted message said:

Quoted message said:

Here's a picture of a real goathead:

home.comcast.netGoathead.jpgOpen ↗

It's sitting on a dime about 0.705" wide.

What kind of a person says "about" 0.705" wide?

I suppose if you describe it to the nearest thousandths of an inch then that's "near enough".

Quoted message said:

Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
question.

[email hidden]

Dear Marty,

Touché!

Actually, a dime is just a hair under 52/73rds of an inch wide.

Alas, my dial calipers don't have fine enough ends to fit in between the itty-bitty ridges, so I
was forced to use Kentucky windage and honorably added the qualifying "about" lest I be accused of
excessive precision.

(Or was joking.)

So how many stickers do ya reckon, plus or minus half a dozen?

Carl Fogel

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