A little help in probability and stats?

[Rain Man]

Man, I'm out of touch on this stuff.

I have 100 orbs*, 50 of which are red and 50 of which are white. I draw 10 of them out randomly without replacement.

How many total combinations can be drawn, and how many of them occur in each of the 10 potential outcomes (e.g., 0 red and 10 white, 1 red and 9 white, etc.)

I need to do this calculation to save the earth from being hit by a large asteroid, so quick help would be appreciated. If we survive, I'll give you rep.





* - I'm not going to say balls because it'll divert the thread.

[RJ]

Train B will arrive in Syracuse before Train A.

[Rain Man]

Sorry for the late response, but I was at a meeting that went until about 10:30 last night.

DanT, that hypergeom.dist function is the perfect tonic for what ails me. I never would've found it myself, but once I figured out how it works, it's really, really nice and easy. Love it. My numbers match austinchief's numbers, so I think I'm good.

The one thing I'm still having trouble with is the number of permutations. I've taken a quick look at DanT's formulae and for some reason I'm not getting Third Eye's numbers. It may be pilot error right now since I'm not familiar with the nomenclature, so I'll work on it a little more. I may have another question later about that part.

[Playful punch to the upper arm] You guys are the greatest! [/Playful punch to the upper arm]

[DanT]

Originally Posted by Rain Man

Sorry for the late response, but I was at a meeting that went until about 10:30 last night.

DanT, that hypergeom.dist function is the perfect tonic for what ails me. I never would've found it myself, but once I figured out how it works, it's really, really nice and easy. Love it. My numbers match austinchief's numbers, so I think I'm good.

The one thing I'm still having trouble with is the number of permutations. I've taken a quick look at DanT's formulae and for some reason I'm not getting Third Eye's numbers. It may be pilot error right now since I'm not familiar with the nomenclature, so I'll work on it a little more. I may have another question later about that part.

[Playful punch to the upper arm] You guys are the greatest! [/Playful punch to the upper arm]

I just noticed a typo in the first line of my longish post, which I've corrected. There are indeed 100*99*...*91 elementary outcomes, which equals 100! / 90!. I incorrectly had typed 100! / 10! in my original post. Sorry about that!

[Rain Man]

Originally Posted by DanT

I just noticed a typo in the first line of my longish post, which I've corrected. There are indeed 100*99*...*91 elementary outcomes, which equals 100! / 90!. I incorrectly had typed 100! / 10! in my original post. Sorry about that!

Oh, that'll make a difference, I bet. I was seeing some really big numbers.

[Rain Man]

Originally Posted by DanT

You good, RainMan?

There are 100*99*98*...*91 = 100! / 90! = 100! / (100-10)! ways to draw 10 balls from 100, denoted P(100,10), if we could distinguish the balls apart. That's the total number of elementary outcomes. Given that we can't distinguish among the white balls or among the red balls, then we can group all these elementary outcomes into 2 ** 10 distinct outcomes where the order would matter.

Here for example, is one of those

WRWRR RRRRW

The above permutation of repeated objects has 3 white orbs and 7 red orbs. Given that I could get 3 white orbs in any of P(50,3) ways from the urn of 50 white balls and I could get 7 red orbs in any of P(50,7) ways, there are

P(50, 3) * P(50, 7 ) ways that I could generate that particular permutation from the elementary outcomes.

So most of this is assuming that we care about which white ball is in which order, right? So there are

50!/47! * 50!/43! = 5.92x10^16 combinations that are possible with 3 white orbs and 7 red orbs, assuming that each orb has a particular identity?

Originally Posted by DanT

Now suppose that I want to group all of the permutations together according to how many unique white balls it has. In other words, I no longer care about the order in which those white ball appear.

Well, each permutation has the same number of elementary possiblities, P(50, 3) * P(50, 7). Note that I could identify a permutation by the location of the white balls. So, for 3 white balls out of 10, there are P(10,3) / 3! distinguishable locations, which we can denote as C(10,3) combinations. So that means that there are C(10, 3) * P(50, 3) * P(50, 7) elementary outcomes that result in a total of 3 white balls out of 10.

If we want to compute the probability of getting 3 white balls out of 10, you just divide the total number of elementary possibilities that yield 3 white balls,

C(10,3) * P(50,3) * P(50, 7) by the total number of elementary possiblities,

P(100, 10 ).

So the number of elementary possibilities if I don't care which white ball or red ball lands somewhere is

100!/90! = 6.29x10^19?

Wait. Shouldn't this be smaller than the number I calculated above?

And in this case, the number of possibilities of a 3/7 split is

50!/47!/3! *50!/47! * 50!/43! = 1.16*10^21?

Wait a minute. I've got something wrong. The second number can't be bigger than the first number. What am I messing up?

Originally Posted by DanT

That's how you get the probability mass function for the hypergeometric. It simplifies very nicely to the formula that is expressed simply using combinations, the formula you'll see in authoritative sources.

I hope you kept that asteroid from hitting us. If you break that rock down a little bit and send it toward the Chiefs front office, I'd appreciate it.

I'm working on it. Astronomically speaking, the front office is rather small so I might have to obliterate most of Kansas City to get it, but it might be a reasonable tradeoff.

[DanT]

Originally Posted by Rain Man

So most of this is assuming that we care about which white ball is in which order, right? So there are

50!/47! * 50!/43! = 5.92x10^16 combinations that are possible with 3 white orbs and 7 red orbs, assuming that each orb has a particular identity?

That calculation would describe how many different ways you could use 50 white balls and 50 red balls to come up with a specific ordered sequence like this one:

WRWRR RRRRW

And there are 120 ordered sequences of length 10 that have 3 white balls in them, so the total number of elementary outcomes that have 3 white balls and 7 red balls is 120 * ( 50! / 47! ) * ( 50! / 43! ) = 7.10 x 10^18 .

In Excel, the formula would be

= combin(10,3) * permut( 50, 3 ) * permut( 50, 43 ) .

Originally Posted by Rain Man

So the number of elementary possibilities if I don't care which white ball or red ball lands somewhere is

100!/90! = 6.29x10^19?

Wait. Shouldn't this be smaller than the number I calculated above?

Actually, it's 6.28x10^19. That's the total number of elementary outcomes. The total number of elementary outcomes should be at least as big than the total number of elementary outcomes that satisfy any particular condition. After all, the total number of elementary outcomes goes into the denominator when computing probabilities, so it needs to be at least as big as what goes into the numerator.

So if we want to compute the probability of getting a "3/7 split", that would be the total number of elementary outcomes with 3 white balls in them divided by the total number of elementary outcomes, which is ( 7.10 * 10^18 )/ ( 6.29 x 10^19 ), about 11.3% . In Excel, the formula would be

=HYPGEOMDIST(3,10,50,100)

[Nickel D]

Originally Posted by AustinChief

I was just looking online for a calc for it.. this is the best I could find http://stattrek.com/online-calculato...geometric.aspx

Should be able to run it for each possible outcome and get your statistics.

(unless I'm not understanding what he is asking for)

Went to that site to crunch the numbers and came up with the final answer of 9.1 million light years.

Hold on a sec...typo found in the site's name: http://startrek.com

Uhmmmmm...nevermind.

[RealSNR]

I need help for a stats seminar. It's an online stats class, basically. The guy who teaches it is cool and definitely respects the tight teaching load so many of us in the course have, but he also gets a bit on the flowery side with his questions.

We're supposed to come up with a response to this:

If light bulb manufacturers labeled their product with descriptive statistics of tested bulb life, what combination of index of center and index of spread would guide your purchase?

So what the **** does "index of center" and "index of spread" mean?

Yes, I've consulted the tutorials and "learning modules" we've been given. I've also Googled the terms, searched around for 2 minutes, and decided to get lazy and see if somebody here knew off the top of their heads.

[Start Croyle]

Originally Posted by SNR

I need help for a stats seminar. It's an online stats class, basically. The guy who teaches it is cool and definitely respects the tight teaching load so many of us in the course have, but he also gets a bit on the flowery side with his questions.

We're supposed to come up with a response to this:

If light bulb manufacturers labeled their product with descriptive statistics of tested bulb life, what combination of index of center and index of spread would guide your purchase?

So what the **** does "index of center" and "index of spread" mean?

Yes, I've consulted the tutorials and "learning modules" we've been given. I've also Googled the terms, searched around for 2 minutes, and decided to get lazy and see if somebody here knew off the top of their heads.

I guess he means mean and variance?

[RealSNR]

Originally Posted by Start Croyle

I guess he means mean and variance?

See, I thought that. Then I got mad because if that's what he meant, he made the problem 1000 times more complicated than he had to.

I'm about ready to assume that's what he means and then send something in