double trouble

[topaz]

Morgan's homework today included this problem (reproduced here verbatim):

If you have one dice with the numbers 1, 2, 3, 4, 5, and 6. Expressed as a fraction, what is the probability of rolling a double (2 of the same number) in 25 rolls?

The probability that at least one number will come up twice is clearly 100%.  We're assuming that they're asking for the probability that at some point two consecutive rolls will turn up the same number.

This seems like a remarkably sophisticated problem to assign a sixth-grader.  It looks like it would have been a reasonable problem for my probability midterm in high school.  Does anyone here disagree?

Edit: The problem was not made up or handwritten by the math teacher -- it was submitted as part of a Math 4 Today handout that he gets assigned on a weekly basis.  For better or for worse, this was part of a standard curriculum math workbook.  (And they wonder is our children learning anything!)

Comments

[rwx.livejournal.com]

this is in line with what i was doing for math in sixth grade. no theory, just how tos.

[matthewwdaly.livejournal.com]

I'm confused by "one dice". If they meant "one die", then you're probably right. I'm wondering if they didn't mean "one pair of dice", which would be a slightly more intuitive problem. I don't think either is a killer as long as you've been taught the addition and multiplication rules for probability.

Still, holy cow. I'm not sure I knew what abstract exponentiation was in sixth grade, and certainly not that numbers as high as 6^25 could be arithmetically processed, and I was a prodigy.

(Sorry for the drive-by.)

[qwrrty.livejournal.com]

Hey, no worries. You're right, if they meant "a pair of dice" by "one dice" then that simplifies the problem a lot. I don't think any of us even considered that interpretation.

[matthewwdaly.livejournal.com]

Your interpretation isn't too rough. For each of the rolls after the first one, the probability of matching the previous roll is 1/6, so the probability that it happens at least once is 1-(5/6)^24. It's a subtle argument that those events are all independent, though, and I'd hate to set a sixth-grader down the path of never thinking about the risk of dependency contaminating the probabilities.

[razil.livejournal.com]

Yes. My first reaction is "0" since if you only have one die you obviously can't roll a double. :)

[buddhagrrl.livejournal.com]

In all seriousness, that was my first reaction too, and I assumed the problem was just incorrectly worded.

[dancingwolfgrrl.livejournal.com]

Me three, although I meant it snarkily, assuming that "one dice" was intended to mean "one pair of dice" and that since they can't type, I am not required to do any math.

[spike.livejournal.com]

I disagree. The standard 'pip pattern' for six is clearly a double-three:

o   o
o   o
o   o

The pip pattern for four is less clearly, but still arguably a double-two. The pip pattern for two is only barely arguably a double-one.

[blk]

I bet this is right, and they meant "a pair of dice." That makes the next sentence make much more sense.

[surrealestate.livejournal.com]

If you have one dice with the numbers 1, 2, 3, 4, 5, and 6.

That's not even a sentence! Plus there's the dice/die issue. There are clearly errors in the text and I'm guessing the actual question is not really expressed as it's written.

[qwrrty.livejournal.com]

Yeah, I know, it's pretty poorly written. I think matthewwdaly may have the right idea.

[surrealestate.livejournal.com]

Indeed, it was probably meant to be "one pair of dice", though that's STILL not a sentence. *shakes fist in the air*

[paradoox.livejournal.com]

It does seem kind of complex and poorly worded. It also seems to be a pain to write (or to figure out without something like Excel).

Assuming I remember what I am doing, a pair of dice 25 times would be the same as 1 die 26 times.

Maybe I just have it wrong.

Or maybe they are looking for 1/1.

[jostajam.livejournal.com]

This is the kind of problem where I write a note underneath saying, "If there is more than one way to interpret the problem, I'm going to pick one for my child and have him solve that. If I pick the wrong interpretation, sorry, we tried, but we're not going to solve for every interpretation of the problem."

[dbang.livejournal.com]

Possible the goal isn't calculation but experimentation?

Anyway, yeah, I got problems much like that in my college Combinatorics class!

[yagagriswold.livejournal.com]

Yup. I was thinking that this reminded me of some of the first problems we attacked in my 300-level probability and statistics class.

[randysmith]

I think it's a poorly worded problem for all the reasons other people have already given, and I'd be annoyed at the school. So the right thing is probably going with jostajam's plan. If they mark him down because you got the wrong interpretation, I'd be tempted to argue with the school about it, but I'm a born windmill-tilter :-J.

from someone who's crap at math but who reads a lot

[queenmomcat.livejournal.com]

Another vote for "it's a badly phrased question"; aside from the fact that they don't specify how many sides the die has/dice have--merely how many numbers--wouldn't having one die result in one answer, multiple dice in quite another? (Yes, I think the question implies two dice, but it's unclear.)

As for whether it's too hard, that depends (as has been said elsewhere) on what he's being taught in math class. I'm not the one to judge whether it's too hard for a sixth grader; I can't do that as an adult.

[rsc.livejournal.com]

Escalating a bit from others' remarks, I'd be seriously disturbed by the idea that a teacher would actually write that first sentence fragment in something to be handed out to students (or, if the teacher didn't write it, that they would hand it out unedited).

[qwrrty.livejournal.com]

Perhaps horrifyingly, it is not something the teacher wrote but something that was printed out from a standardized math workbook.

[jwg.livejournal.com]

I'm impressed by the fact that such a question is asked of a 6th grader. (if course it should have been worded properly). And to me what is important is not the answer but how the student goes about trying to find the answer. In my year teaching 7th grade science (eons ago) I used to ask tough questions and ask the students to explain bow they'd find the answer trying to illustrate how the method used for problem solving is what will help solving other problems.

[dbang.livejournal.com]

I have a bit of a beef with asking young students to explain "how" they got an answer. My 4th graders complain about this incessantly. "Mom, the question is 'what's 18 divided by 9' and they want me to explain how I got the answer. can I write 'duh'?"

Of course I want kids to learn to think things through, but you have to ask questions complex enough that there IS a thought process required. For kids with a strong intuitive grasp of math, asking them to come up with some illustration for their problem solving just slows them down.

On the other hand, my youngest son, who is autistic, often writes down the thought process HE took to get to an answer and it's...well, it isn't a path any non-autistic person would grok. It amuses me to imagine his teachers scratching their head with "huh?"

[qwrrty.livejournal.com]

That makes sense, and the teacher tells us the same thing: she doesn't care so much whether he finishes a particular problem set or does every problem as long as he demonstrates that he understands the principles.

Unfortunately, I don't think that was the intended goal here. Maybe I should have posted a scan of the sheet. It's an 8.5x11 sheet with sixteen problems, printed in a 4x4 grid. There are four square inches of space for each problem, including the text of the question itself. There's no ROOM to explain your reasoning or show your work (a constant frustration when the problem set includes long division puzzles).

Of course he could write more on the back of the page about his reasoning, or use a separate sheet. It just seems like that's not the intent of the exercise here. This all points up some of our ongoing frustrations with the local school and its curriculum.

[cruiser.livejournal.com]

Clearly, the correct answer is supposed to be "Cannot be determined from information provided."