A Maths lesson please!

[peter]

Where am I going wrong here.

If we asign a value of 1 to an event that will happen and 0 to an event that will not happen, then the probability of any event happening or not will be somewhere between 1 and 0.

If good six sided dice is thrown once, the probability of throwing a six is 1/6.

Thrown twice, the probability of getting a six in one of the throws is 2/6 ie. 1/6 + 1/6.

Thrown six times, the probability of throwing a six in one of the throws is 6/6 which equals 1.

1 is the probability of an event that will happen, so if I throw a good dice six times I will throw a six in one of the throws

What has gone wrong here (I bet I'm going to look stupid here )


Also, can anybody tell me if there is a set way of working out the probabilities of random events - say getting knocked over by a bus or looking out of the window and seeing a parrot fly past etc. (You do see these odd sorts of probabilities in the press sometimes so it must be possible, but I'm guessing it's really hard to do.)

[Menolly]

Thrown twice, the probability of getting a six in one of the throws is 2/6 ie. 1/6 + 1/6.

A quick answer from hyperception:

Each throw is considered an independent event.

The probability of a six coming up once is 1/6. The probability of it not coming up is 5/6. After two throws, the probability of getting both sixes is (1/6)^2, but the probability of neither throw being a six is (5/6)^2 or 25/36.

As far as random events are concerned, one would have to know, for example, how many people were looking out windows for what fraction of time versus how many parrots were flying in the vicinity. The numbers that you see in the press are opaque and generally not verifiable. You can ignore them.

[I'm Murrin]

Indeed, your working should be thinking of it as P = 1 - NP, where NP = (5/6^n), n being the number of throws.

When you just want the chance of at least one roll being a six, it's the inverse of the chance of none of them being a six.

Each potential combination of rolls is defined by multiplying the chances on each roll, and you sum together all the probabilities for all the combinations you want (i.e., the chance of only one six (not 0, 2, or 3) in three rolls would be 1/6*5/6*5/6 + 5/6*1/6*5/6 + 5/6*5/6*1/6). In this case you want the chance of any sixes at all, which is all possible combinations (the sum of which is 1) minus the one combination where you get no sixes.

[wayfriend]

Yep. You multiply probabilities, not add them.

Otherwise, rolling the dice 7 times would have a 1.167 probability of getting a six. Which is plainly invalid.

As for the probabilities of "random events" (busses, parrots, etc.) no one can tell you the probability unless they have some data and some assumptions, even if they are estimates.

For example, if we know the number of parrots per square mile, the percentage of time a parrot spends flying, assume a uniform distribution of flying parrots, and measure the area in which a flying parrot is visible from the window, you could come up with a probability of seeing a flying parrot out ones window.

And then you might discover other factors come into play - do parrots avoid windows? Would you notice a light blue parrot flying in the sky? Does approaching a window scare parrots into flight?

As you can see, that's a lot of data to gather.

[High Lord Tolkien]

Thrown twice, the probability of getting a six in one of the throws is 2/6 ie. 1/6 + 1/6.

A quick answer from hyperception:

Each throw is considered an independent event.

Yep.
Took me a while to figure this one out too.
I was told once that the chances of something (skipping the details here) bad happening during my wife's pregnancy was 1 in 5,000 and I was freaking out because at this hospital they do around 5000 pregnancies a year. I was thinking that it was definitely going to happen to one mother that year....

[DukkhaWaynhim]

Murrin's equation is correct.

Probability is not certainty. It is a measurement of likelihood, so no numerical representation of a die throw, or any random event, should ever be 0 or 1, unless you are foxing the randomness somehow.

Each time you throw the die, you have a 1/6 chance of rolling a 6. Rolling a certain large # of times increases the calculated overall probability that at least one of those throws will be a six, but you can never be 100% certain that X throws will result in a six. This is true no matter how many times you throw the die.

dw

[Fist and Faith]

Peter, there's a very nice, visual explanation here:
www.edcollins.com/backgammon/diceprob.htm
The odds of rolling a 1 on either either of the throws is 11 out of 36, or 30.5%.

The chance of getting a 1 if you rolled it six times is 31,031 out of 46,656, or 66.51%.

It's an odd thing, eh? You can roll a thousand dice at once, and you might not get a 1 on any of them. Unlikely, but the possibility is not 0.

And the chances of two people in a random group of 8 having the same birthday is based on the fact that there are 28 different pairings.

And an infinite number of monkeys typing on an infinite number of computers forever might NOT ever type the complete works of Shakespeare. Heck, it's possible they'd all type "e" forever. Or maybe they'd never type "e".

[Orlion]

Odd, I thought if you managed to throw the dice an infinite number of times, you would get a six, guaranteed (provided certain assumptions about the dice... like it having a six). You now, kinda a limit thing. As x approaches infinity, f(x) [in this case 1/x] approaches zero.

I say this because I read in a Scientific American once that the idea about multiple universes is that with random infinite number of universes, you'd get all the possibilities. For example, life developing on Earth may have had a 1/10^100000000000000...... chance, but if there's an infinite number of universes, it will happen. Hell, it'll happen an infinite amount of times! It'll just be infinitely smaller than the infinite amount of times life didn't develop on Earth.

I was hoping to fit more infinites in there...

[Vraith]

Odd, I thought if you managed to throw the dice an infinite number of times, you would get a six, guaranteed (provided certain assumptions about the dice... like it having a six). You now, kinda a limit thing. As x approaches infinity, f(x) [in this case 1/x] approaches zero.

I say this because I read in a Scientific American once that the idea about multiple universes is that with random infinite number of universes, you'd get all the possibilities. For example, life developing on Earth may have had a 1/10^100000000000000...... chance, but if there's an infinite number of universes, it will happen. Hell, it'll happen an infinite amount of times! It'll just be infinitely smaller than the infinite amount of times life didn't develop on Earth.

I was hoping to fit more infinites in there...

heh...the odds approach zero, but I believe they never actually reach zero...and it goes in both directions, I think...there is also a non-zero chance that every toss would be a 6.

The thing is, though, as a practical matter I doubt either can ever really happen. Because the math is pure and assumes "All other things being equal." In the real world all other things are not equal.

On the universes, the Odds are that will happen a smaller infinity times...but a non-zero chance it will never happen [actually USED to be a non-zero...now it's happened, we rolled a 6!], and a non-zero it will happen every single time...but unlike dice, we don't actually know the odds of life in a universe. Maybe it is 1/1, every universe evolves it. Hell, maybe it's a billion over 1, every universe evolves life at least once, but a billion different times, on average.

[Obi-Wan Nihilo]

I think a lot of these apparently probabilistic scenarios like throwing dice ad infinitum are actually more deterministic than a simplistic abstraction would suggest. In fact it is difficult to imagine purely probabilistic events apart from, say, quantum physics.

[peter]

Thanks guys. I think I'm getting where I was going wrong (not absolutely sure since I did read somewhere that all events could be placed on a probability gradient of 1 (ie they happen) to 0 (ie they don't). Clearly from fist's post the probability of getting the 6 at one throw is going to be 66.whatever up toward the 1 end if the scale (.66ish? - I would have thought it would have been more, but there you have it). My mistake is clearly that the probabilities of individual events are not additive in the way I supposed. Oh well - back to the drawing board I guess

Tell me is probability a branch of statistics though. I heard the following funny little story on the misleading nature of stats the other day (hope I can get it right!) and if probability sits inside this discipline I doubt I'll ever get a handle on it.

There is a fatal genetic condition that affects one in a thousand people and a test is developed that is 99% accurate in diagnosing it. A guy goes for the test and gets a positive result and naturally is down in the dumps. The doctor reassures him and the guy remonstrates "But I'm going to die! A 99% accurate test has shown me to have the condition".
"Well" says the Doc, "Lets have a look at that". "If I test 1000 people, one of which is likely to have this condition, the 99% accuracy of the test means that I will get ten false positives plus the one true positive ie one false for each 100 people tested. Therefor even with a 99% accurate test your chances of developing the condition are actually only 1 in 11. I'd gamble on those odds."

The moral of this story is 'Never forget the base line - thats where the real import lies'.

This seems on the face to be a trivial little story, but on one occasion a good friend of mine was told on the basis of an early blood test that there was a forty% chance the baby she was carrying was Down's syndrome. She was offered an immediate termination or the choice to wait untill the definative test for Down's, an amniocentesis, could be carried out in a few weeks time. It was not my place to influence her and she elected for the early termination, but she clearly had no understanding of what the forty% result might or might not have meant and I often wonder to this day what might have been the result if she had chosen to wait for the second, definative test.

[DukkhaWaynhim]

By its denotation, I believe that probability calculations are part of statistics as a course of study, but I personally think of probability as being derived from statistics, since probability is really being used to attempt to predict future statistics.
Statistics are measurements of actuals. Probability is a prediction of possibles to make statments of likelihood.

I am completely thrown by your doctor/genetic disease example, because as soon as I hear that the test is 99% accurate, I want to ask more about the test itself - when it fails, does it result in false positives, false negatives, or both? Plus, the statement that 1 in a 1000 develop the condition is a general statement of rarity, but until you are accurately diagnosed with the condition, you don't know whether you belong in the 999 or the 1. A statistic is a mathematical tool -- people in the world aren't walking around in discrete packs of 1000, where 1 in that crowd is always bearing that condition. You don't get to set your population, so a randomly chosen population *could* be 1000 people with that rare condition -- while that is highly unlikely, it is still possible.

dw

[aTOMiC]

One thing to consider in these calculations it the "luck" factor which increases proportionally with the urgency of the situation.
For example if you roll a dice to decide which of six different locations you are considering for dinner the odds of you rolling the corresponding number that represents your favorite is fairly even.
In an example where the stakes are higher (a roll of a dice to determine whether you will receive a billion dollar tax free check delivered personally by the president) you have a one in one hundred billion chance of rolling the hoped for number.

I'm just sayin'.

[Avatar]

Yeah, it's part of statistics. At least, I studied it as part of the mandatory Statistical Methods class I had to take at uni.

(Man, did I hate that class. It was a requirement because I had to study research methodology for doing a case study for one of my majors.)

(I can't remember much about it any more though. One thing I've definitely never used. )

--A