A spider eats three flies a day. Until she has done this, she has an even chance of catching a fly that flies through her web. A fly is about to fly through the web. What is the fly’s chance of survival, given that today five flies have already flown through his web?

If you find this problem challenging, try it.

More below the fold: —> If you find this problem interesting, but not a real problem for you, here’s a challenge:

1. Guess an approximate grade level that this problem is appropriate for, and
2. Write a new problem, based on this one, but for a more advanced student, and then
3. Write a new problem, based on this one, but for a less advanced student.
4. Can you alter this one to create a nice problem to challenge high school math teachers?
February 21, 2007 am28 7:54 am 7:54 am

Well, I think the most natural hardening is, “Given that today n flies have flown through the web, what is our fly’s chance of survival?”

But there are other variations. I’m not sure which of these are harder and which are easier. For instance, “Given that our fly has a 5/16 chance of survival, how many flies have flown through the web today?”

Or, we can play around in other ways: “If 3 (or 10, or n) flies will fly through the web today, what is the probability that the spider gets enough to eat?”

“If 3 (or 10, or n) flies will fly through the web today, what is the expected number of flies the spider eats?”

Here’s my version to challenge math teachers (although maybe it’s not the hardest one I’ve tossed out):
“Assuming that flies will keep coming all day until the spider eats its fill: if the $m$th fly is the third one the spider eats, what is the expected value of $m$?

The grade level for your question? I think bright middle-school students who’ve learned some probability could get it, maybe with a little struggle. I would guess that most competent 10th graders could hammer it out, in principle. I’m probably not a good estimator of this kind of thing, though.

2. February 21, 2007 am28 8:00 am 8:00 am

using LaTeX for italicized variables, are we? Just too cool.

I like the expected value variation. It will likely be harder to come up with a good easier problem. (Easier’s not so hard, but making it good…)

February 21, 2007 am28 9:58 am 9:58 am

It seems to me the hardest part about this is that it’s easiest if you’ve studied probability and can invoke the right formulae… I don’t think I’d have seen this formally until I was in college.

Let $P_S = {\rm the probability the fly survives}$.
Let $P_3 = {\rm probability the spider has eaten 3 flies already}$, which I think is 0.5.

$P_S = P_3 \times 1 + (1-P_3) \times 0.5 = 0.75$

If I were going to make it easier, I’d say find the probability of survival if 3 flies had flown into/through the web already — that would focus the challenge on combining the “hungry spider” and the “not hungry spider” cases, and not on the calculating the probability that if 5 flies had flown by already the spider was still hungry.

Now let’s see if all that latex worked…

February 21, 2007 am28 10:03 am 10:03 am

Trying the invalid formula once more…

$P_3 = {\rm probability~spider~not~hungry}$

5. February 21, 2007 pm28 10:32 pm 10:32 pm

So, I would think about it like this,

figure out the total number of ways that 1 fly could have been eaten already, the number of ways that 2 flies could have been eaten already, and 3 and so forth…

the sum of the possible ways that either 1 or 2 flies could have been eaten, divided by the total number of possibilities, gives the chance that the spider is still hungry…

then multiply by 1/2…?

6. February 21, 2007 pm28 10:48 pm 10:48 pm

I forgot to include the possibility that the spider hasn’t caught a d*mn thing…

7. February 21, 2007 pm28 10:59 pm 10:59 pm

Sorry to be annoying and post three in a row, but I also forgot to add in the probability that the spider is sated, times 1… the comments above reminded me of that.

February 22, 2007 am28 12:31 am 12:31 am

I agree with rdt, although s/he didn’t supply any of the calculations that kelly was working with (which sounded like they were headed down the right path, but just not finished off).

Answering one of my own questions: “If $n$ flies will fly through the web today, what is the expected number of flies the spider eats?”
There is only 1 (=${n \choose 0}$) ways the spider can eat 0 flies, so this contributes an expectation $\frac{1}{2^n}\cdot0$. There are $n = {n \choose 1}$ ways the spider can eat 1 fly, so this contributes an expectation of $\frac{n}{2^n}\cdot 1$. There are $\frac{n(n-1)}{2} = {n\choose 2}$ ways the spider can eat 2 flies, so this contributes an expectation of $\frac{n(n - 1)}{2^{n + 1}}\cdot 2$. In all other circumstances, the spider eats 3 flies, so this contributes an expectation of $(1 - \frac{1}{2^n} - \frac{n}{2^n} - \frac{n(n-1)}{2^{n+1}})\cdot 3$. We then add these four values to get our total expectation, $3 - \frac{n^2 + 3n + 6}{2^{n+1}}$.

9. February 22, 2007 am28 12:43 am 12:43 am

Clarification:

Combinations of n things taken r at a time =
C(n,r) =
nCr = (the n and r should be subscripted. Anyone help me with the html or latex?)
$\binom{n}{r}$

JBL’s notation is what we expect in college, the middle one in high school, the first one, well, that’s the one I don’t understand why we don’t see more of.

February 22, 2007 am28 12:51 am 12:51 am

_nC_r gives subscripts. You need to use brackets it you want more than one character, e.g. _{11}C_6. Otherwise you get ugliness like $_11C_6$.

11. February 26, 2007 am28 7:28 am 7:28 am

So what grade level is that?

by Ogden Nash

Footnote:

A Flea and a Fly in a Flue

A flea and a fly in a flue
Were imprisoned, so what could they do?
Said the fly, “let us flee!”
“Let us fly!” said the flea.
So they flew through a flaw in the flue.

12. February 26, 2007 am28 7:50 am 7:50 am

Ogden Nash? I memorized the Elephone and the Man who Wasn’t there in early grade school, but I recite them (and the Codfish, and the chair that wasn’t there, and some others) to high school students, and none of them have ever heard any of them.

The algebra is not hard. The conditional probability is not taught in most of New York – so an advanced senior. With a teacher guiding an off-topic problem solving group, it could be handled in 9th or 10th grade, maybe using a complete list.

I was fortunate enough to go to a school (another state) where we learned serious combinatorial probability in 11th grade precalc. That’s when I would have been able to handle this one.

- that wriggled and wriggled and wriggled inside her… (traditional)