Rally Apr. 17th-19th

[gd_barnes]

Originally Posted by mdettweiler

Somewhat similar situation here--for about a year or so I've had one Core 2 Duo which produced many of my primes, and then a few months back I added a Q6600. The Q6600 alone has produced plenty of primes; I've found that at current n-levels, it can crank out about 1000 k/n pairs per day on port 8000. Gary, what's the odds of finding a prime in 2000 k/n pairs on the 10th Drive at n~515K? I remember it was a bit less than 100% for n=400K, but I'm not sure about >500K.

The chances of something happening are almost never "a bit" less than 100% unless you run about 5 times more tests than are the chances of that thing happening. Example: On a sieve to P=20T at n=400K, to have a 99% chance of finding a prime, you'd have to run nearly 23,500 tests! For a 95% chance, you'd need nearly 15,300 tests.

I think what you mean is that you should find, on average, 1 prime at n=400K in 2000 tests. Regardless, your memory fails you. That's way too optimistic. No, it's not near that frequently. The odds of finding a prime at n=400K are as follows:

Sieve to P=4T like the mini-drive: One test: 1 in 5365; 2000 tests: 1 in 3.2
Sieve to P=20T like the 10th drive: One test: 1 in 5083; 2000 tests: 1 in 3.1

I gave Gamerz the odds of prime on the mini-drive at n=422K.

The 10th drive is now near n=520K. At that level, the odds of prime on the sieve to P=20T are:
1 test: 1 in 6608
2000 tests: 1 in 3.8 or 26%
6608 tests: 1 in 1.6 or 63%

By the time the rally starts, we'll probably be at n=530K or higher and the rally will probably take us near or past n=540K by the end of it so it will be a little less than the above during the rally.

Attached is an Excel spreadsheet that I used to come up with these odds. It's fun to play with. Most of the formulas were given to me by AXN1 on the forums although I came up with a few myself such as the ones for twins, etc.


Gary

[mdettweiler]

Originally Posted by gd_barnes

The chances of something happening are almost never "a bit" less than 100% unless you run about 5 times more tests than are the chances of that thing happening. Example: On a sieve to P=20T at n=400K, to have a 99% chance of finding a prime, you'd have to run nearly 23,500 tests! For a 95% chance, you'd need nearly 15,300 tests.

I think what you mean is that you should find, on average, 1 prime at n=400K in 2000 tests. Regardless, your memory fails you. That's way too optimistic. No, it's not near that frequently. The odds of finding a prime at n=400K are as follows:

Sieve to P=4T like the mini-drive: One test: 1 in 5365; 2000 tests: 1 in 3.2
Sieve to P=20T like the 10th drive: One test: 1 in 5083; 2000 tests: 1 in 3.1

I gave Gamerz the odds of prime on the mini-drive at n=422K.

The 10th drive is now near n=520K. At that level, the odds of prime on the sieve to P=20T are:
1 test: 1 in 6608
2000 tests: 1 in 3.8 or 26%
6608 tests: 1 in 1.6 or 63%

By the time the rally starts, we'll probably be at n=530K or higher and the rally will probably take us near or past n=540K by the end of it so it will be a little less than the above during the rally.

Attached is an Excel spreadsheet that I used to come up with these odds. It's fun to play with. Most of the formulas were given to me by AXN1 on the forums although I came up with a few myself such as the ones for twins, etc.


Gary

Hmm...I see. I apparently have been somewhat mistaken as to the definition of the term "x in y odds"--I thought that if something has a 1 in 4 odds, then it's 1/4=25% and thus doing 4 runs means that it should on average produce at least one positive (assuming a random distribution). Is this incorrect?

[IronBits]

I've started up all I7 cores on Port8000 until I receive the care package for sieving.
Not really Rally related, but, I wish it was counted towards the Rally.

[gd_barnes]

Originally Posted by mdettweiler

Hmm...I see. I apparently have been somewhat mistaken as to the definition of the term "x in y odds"--I thought that if something has a 1 in 4 odds, then it's 1/4=25% and thus doing 4 runs means that it should on average produce at least one positive (assuming a random distribution). Is this incorrect?

Please reread my post all the way through (and this one too). Obviously you have missed what I was attempting to imply.

Where does the 1 in 4 odds come from? No, believe-it-or-not, if something has a 1 in 4 chance of happening AT LEAST one time, if you do it 4 times, you should, on average, have it happen MORE THAN 1 time. What you're missing is the term "AT LEAST". If there is a 25% chance of it happening at least one time, occassionally it will happen 2, or even 3 or more times.

Also, it does not have "a bit" less than 100% chance of happening if you run it 4 times like you stated in your original post. It only has about a 62% chance of happening. You have to account for the fact that it can happen MORE than one time, which reduces the chance that it will happen one or more times.

62% is the "golden ratio" minus 1. See http://en.wikipedia.org/wiki/Golden_ratio. It is the chance of something happening if you run it the same # of times as its chance of happening in one trial. Example:

Pick a number on a standard 38-number roulette wheel. Now spin that wheel 38 times. There is a 62% chance that the # will come up at least one time in those 38 spins. To average out to 1 hit, it will have to sometimes hit 2 times and sometimes even 3 or more.

The same applies to prime numbers. If there is a 1 in 4000 chance of a prime and you run 4000 tests, you have a 62% chance of AT LEAST one prime coming up, not a "bit less" than 100%.

Something more interesting: If there is a 1 in 4000 chance of a prime, if you run 1000 tests, you will NOT have a 1 in 4 chance of finding a prime number for the same reason that you will not have a 100% chance of finding a prime in 4000 tests. The odds have to account for the fact that there can be 2 or more prime numbers in your test such that the AVERAGE over the long run is 1/4th or 0.25 primes in 1000 tests.

The most remarkable thing though: For a 1 in 4000 chance of prime, if you run 1000 tests, the odds are only ~22% of finding one or more primes even though the long-term average is 0.25. If you run 2000 tests, the odds are ~39% of finding one or more primes even though the long-term average is 0.5.

What you're missing is that in order to average out to 0.25 primes in 1000 tests, you have to take into account the fact that you will sometimes hit 2 or more primes in that stretch. The long-term average will still be 0.25; it's just the chances of finding AT LEAST one prime in 1000 tests are less than 1 in 4.

2 final things:
(1) For a 1 in 4000 chance of prime, to have a 50% chance of finding at least one prime, you have to run ~2775 tests.
(2) For a 1 in 4000 chance of prime, to have a 25% chance of finding at least one prime, you have to run ~1150 tests. But here is the kicker: If you run ~1150 tests 4 times, you will, on average, get 1.15 primes because you will have run ~4600 tests. As stated in the 1st para., it's more than 1 prime!

Confused yet? :-) Probability and statistics are anything but intuitive unless you study them more in depth. Math is fun. Take the spreadsheet and fiddle around with it. That will help.


Gary

[Flatlander]

My brain hurts.

Can you please formulate a proof to show that it is impossible for my wife to change her blouse without first removing her sweater. She keeps doing it and it's really annoying.

[IronBits]

What does any of that have to do with Rally Apr. 17th-19th ?

[Mini-Geek]

Originally Posted by IronBits

What does any of that have to do with Rally Apr. 17th-19th ?

Well, it makes a wee bit more sense when you see how the topic changed. The split began around #8 and was complete by #12. (talking about not having many cores to bring to the rally, which led to the odds of finding a prime with said, or similar, small resources)

Originally Posted by Flatlander

My brain hurts.

I second this recommendation:

Originally Posted by gd_barnes

Math is fun. Take the spreadsheet and fiddle around with it. That will help.

It's much more intuitive and fun when you're playing around with it than reading a bunch of numbers in Gary's post.

[gd_barnes]

Originally Posted by Mini-Geek

Well, it makes a wee bit more sense when you see how the topic changed. The split began around #8 and was complete by #12. (talking about not having many cores to bring to the rally, which led to the odds of finding a prime with said, or similar, small resources)
I second this recommendation:
It's much more intuitive and fun when you're playing around with it than reading a bunch of numbers in Gary's post.

LMAO. Oh, come now, like Mini hasn't posted a few odds of his own in a thread or 2 here.

That said, yes, math is much more fun then reading my annoying and rambling posts. I just get into the odds of stuff so much that my enthusiasm rambles a little.

Chris, my nearly 14-year-old daughter, when I'm taking her from a soccer game to soccer refereeing sometimes will change her top in the car on the way. She leaves her soccer jersey on top while pulling off her tshirt underneath and then putting on her referee jersey underneath. She then takes off her soccer jersey and she's all done! :-) If I look over at it her, she just grins real big like she's being mischevious. I think it's hilarious. I've witnessed some of her friends do the same thing right before a game when coming from some other activity.

Therefore I have a theory: It is possible for a woman to change her top underneath an outer garment. I haven't formulated a mathematical proof yet but I'm sure there is one.


Gary

[Mini-Geek]

Originally Posted by gd_barnes

LMAO. Oh, come now, like Mini hasn't posted a few odds of his own in a thread or 2 here.

I know, but that doesn't mean I think my posts about odds are more interesting than the math.

Originally Posted by gd_barnes

Chris, my nearly 14-year-old daughter, when I'm taking her from a soccer game to soccer refereeing sometimes will change her top in the car on the way. She leaves her soccer jersey on top while pulling off her tshirt underneath and then putting on her referee jersey underneath. She then takes off her soccer jersey and she's all done! :-) If I look over at it her, she just grins real big like she's being mischevious. I think it's hilarious. I've witnessed some of her friends do the same thing right before a game when coming from some other activity.

Therefore I have a theory: It is possible for a woman to change her top underneath an outer garment. I haven't formulated a mathematical proof yet but I'm sure there is one.

Hard to imagine how that's possible. Any idea?

[gd_barnes]

Originally Posted by Mini-Geek

I know, but that doesn't mean I think my posts about odds are more interesting than the math.

Hard to imagine how that's possible. Any idea?

Point taken on the 1st part. Although I never claimed my posts were more interesting than the math, perhaps I implied it. (lol)

On the 2nd one, it's tricky but not difficult once you've done it a couple of times. She has to slip her arms out of both garments while leaving both garments on her body with both arms underneath both of them. Once that is done, then it's easy enough to pull the undergarment off after slipping it up over her head (under the outer garment) and then pulling it out of one of the arm holes or the head hole of the outer garment.

As for putting the new garment back on, actually, I think I misstated. I had originally said that she put it on under the existing one and then pulled the old one off but after thinking it through, I remember it better. What she actually does is put the new garment on OVER the one that is already on and then pull the "now" undergarment off the same way she did the original under-garment. In other words, during the whole ordeal, she doesn't have her arms in any of the arm holes until the very end when she puts them through the final garment.

Now, as for Chris's wife changing blouses completely underneath her sweater: That is more difficult and is what I had implied originally. It would require putting on the new blouse underneath the sweater in the reverse manner that she pulled off the original one.

I've tried it the way Amy (daughter) does it with a regular shirt and tshirt a couple of times and it takes a time or 2 to get used to it but it's not hard once you've done it a couple of times.

Now the only question is: Can they do it with their pants? Chris, ask your wife to do that. I'm pretty sure it would be impossible with a standard pants design. With a shirt, you can pull your arms out and keep it on your body. With pants, you can't pull your legs out and still keep them on your body.


Gary

[IronBits]

They can if it's one of them pants/skirt things.

I'm bringing quite a few cores, how many are the rest of you bringing to the Rally?
Are you going to let Free-DC slam dunk you again ????