Wikipedia:Reference desk/Archives/Mathematics/2008 November 10

= November 10 =

probability
I'm having trouble figuring out the following problem.

You have gone fishing and have caught five fish. Two of these fish are below the legal size. The Fish and Game Warden stops you and checks your catch by pulling out two fish at random from your creel. What is the probability you will be caught? (He does not return the fish before he pulls out the second) I thought this would be 2/5+2/4 but that can't be right. What is the method to answer this question.

Thanks--74.129.0.221 (talk) 02:11, 10 November 2008 (UTC)

GOING CRAZY OVER THIS PROBLEM...PLEASE HELP!!!--74.129.0.221 (talk) 02:36, 10 November 2008 (UTC)

OK OK, I think I got it, you have to look at the chance he doesn't get caught 3/5 times 2/4 = 30% subtract that from 100 and you get 70%. THANKS FOR NOTHIN MATH NERDS.--74.129.0.221 (talk) 02:53, 10 November 2008 (UTC)


 * You're welcome. And thank you for noticing that the people who answer questions on this page are all volunteers, and make no promise that your question will be answered within, say, a 45-minute period. Have a nice day. Confusing Manifestation (Say hi!) 03:07, 10 November 2008 (UTC)


 * And if you feel like giving something other than contempt, have a go at writing an acceptance sampling article, now that you've grasped the main point.…86.152.78.153 (talk) 15:26, 10 November 2008 (UTC)

Just a joke guys, thanks anyways, no harm intended, much love for wikipedia.--74.129.0.221 (talk) 18:48, 10 November 2008 (UTC)

Say you have n fish, p bad fish out these n fish, and an inspector chooses k fish (out of these n fish). What is the probability that you will be caught?

Answer: Hypergeometric distribution

Topology Expert (talk) 08:15, 11 November 2008 (UTC)


 * From a quick look at the article, I think not. Shouldn't it be $$1 - {{{n-p} \choose {k}} \over {n \choose k}}$$ ? —Tamfang (talk) 07:04, 12 November 2008 (UTC)

Out of a total of $$\tbinom nk$$ possible samples, $$\tbinom{n-p}{k-i}\tbinom p i$$ samples contains $$i$$ undersized fish. The probability of getting $$i$$ undersized is $${\tbinom{n-p}{k-i}\tbinom p i}/{\tbinom nk}$$.

Tamfang's result follows from $$P(i\ne 0)=1-P(i=0)$$, and 74.129.0.221's result follows from $$n=5, k=2, p=2$$:
 * $$1 - \tbinom{5-2} 2 \tbinom 2 0/\tbinom 5 2=1-3\cdot 1 / 10=0.7.$$ Bo Jacoby (talk) 09:14, 12 November 2008 (UTC).

Another interesting (but simple) problem is this:

If I have n fish and k of them are over-size, what is the probability that I need to pick out l (l < n) of the fish to obtain precisely p (p < min{l, k}) over-size fish?

Answer: Very similar to a hypergeometric distribution but not the same. Try to find this out yourself (try not to see the hint).

Hint: In the picking of the first l-1 fish, I need to have p-1 oversize fish and in the last pick I need to have another oversize fish. Use the hypergeometric distribution for the first l-1 picks.

Topology Expert (talk) 09:36, 12 November 2008 (UTC)

Driving carefully
You might like this little puzzle. This is what I've told some children before driving them:


 * The roads are full of idiots driving recklessly at high speed.
 * The longer you drive the more chance there is of meeting one of these idiots and having an accident.
 * I'm a good driver, I like to drive safely and reduce the chances of an accident
 * So I'll be driving as fast as possible so we get off the road quickly and not risk having a crash

For some reason or other they don't seem to follow my logic. Do you see anything wrong with it? :) Are there circumstances where its equivalent is true? I've actually come across almost its equivalent in another context where a person just wasn't able to balance up two risks. Dmcq (talk) 16:40, 10 November 2008 (UTC)


 * You'll find lots of similar examples in game theory. I can't offhand think of an exact analog of your scenario, but the general idea that independent maximization of personal welfare ends up reducing everyone's welfare has long been known as the tragedy of the commons.  —Ilmari Karonen (talk) 17:20, 10 November 2008 (UTC)
 * This isn't the tragedy of the commons, though. This is the fallacy of treating your welfare as a function of one variable (in this case how many foolish high-speed drivers you meet) and ignoring all others. Algebraist 17:30, 10 November 2008 (UTC)


 * You seem to have increased the probability that other drivers will encounter an idiot driving recklessly at high speed. APL (talk) 17:23, 10 November 2008 (UTC)

It sounds like you're silently assuming driving fast isn't reckless. If this is true, consider this analogy: "There are a lot of bad drivers on the road, so we will fly in an airplane, statistically much safer, straight over them". Then add the condition "if there's an accident in a part of the road JUST as we're over, we consider it as our accident". In that case it's better to fly as fast as possible, to reduce your time over the road! —Preceding unsigned comment added by 82.124.214.224 (talk) 17:36, 10 November 2008 (UTC)


 * The logic does seem to imply though that the better the driver I am the more other people will contribute to accidents, so the faster I should drive. In the limit I should just floor the accelerator if I'm really good. Dmcq (talk) 18:29, 10 November 2008 (UTC)
 * In the limit where you're a perfect driver, and can forsee all possible problems long beforehand with your psychic powers, then sure, why not drive as fast as possible? For non-perfect drivers without psychic powers, there comes a point at which going faster increases danger. Also, other people causing you to have accidents is not independent of your speed: that idiot pulling out in front of you might be easier to dodge if you weren't driving at 200mph. Algebraist 18:32, 10 November 2008 (UTC)

There's a sort of blind tee-junction near where I live (UK; we drive on the left) where I often have to take a right turn. My considered opinion is that it's better to turn right quickly to reduce the time spent at risk of being hit. After all, if I wanted to get hit, I would turn vvvveeeerrrrryyyy slowly and dawdle in that dangerous bit of road, and hope that some nutter [or indeed, perfectly sensible driver] would pile in to me. just my 2c. Robinh 21:50, 10 November 2008 (UTC)
 * Well, yes, being in the middle of the road pointing in the wrong direction for as little time as possible sounds wise. --Tango (talk) 22:24, 10 November 2008 (UTC)
 * If it's completely blind from the other side, with no reaction time, I think you're wise. That's why the bad guys move very quickly once their watch yells "Coast clear!"  —Preceding unsigned comment added by 83.199.126.76 (talk) 22:36, 13 November 2008 (UTC)

A slightly better model gives a sensible result. The accident intensity has one contribution proportional to time, and another contribution proportional to speed. The total accident intensity is minimum when the speed is chosen such that the two contributions balance. (Because d(av&minus;1+bv)/dv = &minus;av&minus;2+b = 0 when av&minus;1 = bv). The philosophy of the golden mean has this mathematical interpretation. You were neglecting the accident intensity contribution from speed. Bo Jacoby (talk) 09:44, 11 November 2008 (UTC).
 * My accidents per unit time would have to go up more than proportionally with speed otherwise the equation would be (a+bv)/v and differentiating that gives av&minus;2 = 0 or v = ∞. Interesting about coming out at a T-junction though. That's an exact correspondence, I'd been looking through the games section for something similar. Sounds very hairy though! I think I'd try a different route if at all possible. Dmcq (talk) 16:37, 11 November 2008 (UTC)
 * Very nice little puzzle with a Moebius-band flavour in its paradoxical way of revesing the hypothesis... but it is perhalps worth warning all idiots that may be reading: do not try it in a real road!--PMajer (talk) 20:14, 11 November 2008 (UTC)

The velocity dependent contribution to the average cost per kilometer due to accidents is roughly proportional to the kinetic energy of a collision, E&middot;dt/dx = (mv2/2)&middot;v&minus;1 = bv. Bo Jacoby (talk) 20:41, 11 November 2008 (UTC).
 * The damage caused by a collision is, but what about the chance of a collision? What does that have to do with kinetic energy? --Tango (talk) 20:43, 11 November 2008 (UTC)

Perhaps the logic of driving with infinite speed is correct if you are only interested in minimizing the number of collisions and don't care about the damage done by collisions. I once had an accident to my car when it was parked (v=0) and a drunk pedestrian stumbled into it. The damage was limited but the collision count was one. Bo Jacoby (talk) 20:55, 11 November 2008 (UTC).
 * Indeed. If, for example, you attempt to drive from your house to the supermarket without ever travelling slower than, say, 500 km/h, it's a quite safe bet that you will only ever have one car accident in your entire life. Maelin (Talk | Contribs) 13:28, 12 November 2008 (UTC)
 * Of course, damage isn't an exact science. I mean, generally a crash at 30mph is going to do more damage than one at 10mph, but it depends where you hit. I was in a crash about 5-6 years ago. It was on a campus road, we were doing 20mph or so when someone came out of a junction without looking. Swerving and brakes probably reduced speed to about 10mph collision. The other car hit the side of our car, just behind the front wheel. That burst the radiator, caused several thousand pounds worth of damage and the car was a write-off. The other car drove away with nothing more than a bent numberplate and a bonnet which wouldn't open. Same collision, so same impact speed, but radically different costs. -mattbuck (Talk) 20:37, 12 November 2008 (UTC)