Math Geeks? Can You Solve this? Seems Easy, But I've No Clue.

[SGOS]

I've came up with a math problem after applying weed killer to my lawn, and the directions say it should be watered a day or two later.  So I looked up the weather forecast, and it said that the chance of rain for the following three days is 50%, 40%, and 30%, so what is the chance that it will rain at some point over the entire 3 day period?  It seems like maybe you could average the three percentages, but I don't think that works, because with tomorrow alone there is a 50% chance, so for the three day period there has to be a minimum of a 50%.  Day two is less than 50%, but it seems like adding another day, even if it is a lower chance, should actually increase the chances over all.  Is there an answer here or not?

Put another way if there is a 10% chance for 100 days in a row, wouldn't a chance of rain over that many days approach a certainty?

[Baruch]

I've came up with a math problem after applying weed killer to my lawn, and the directions say it should be watered a day or two later.  So I looked up the weather forecast, and it said that the chance of rain for the following three days is 50%, 40%, and 30%, so what is the chance that it will rain at some point over the entire 3 day period?  It seems like maybe you could average the three percentages, but I don't think that works, because with tomorrow alone there is a 50% chance, so for the three day period there has to be a minimum of a 50%.  Day two is less than 50%, but it seems like adding another day, even if it is a lower chance, should actually increase the chances over all.  Is there an answer here or not?

Put another way if there is a 10% chance for 100 days in a row, wouldn't a chance of rain over that many days approach a certainty?

The problem is ... is the odds of it raining day 2, dependent on if it rains on day 1 ... or is the odds of it raining on day 3, dependent on if it rains on either or both day 1 and day 2?

If the odds are independent day by day ... then the odds of it not raining (easier to calculate) is (how to calculate the odds of flipping heads three times in a row would be 0.5^3) 0.7x0.6x0.5 or 0.168 chance that it won't rain on any of the days ... so the odds of it raining on at least one of the days is 1.0-0.168 or 0.832

[Hakurei Reimu]

I've came up with a math problem after applying weed killer to my lawn, and the directions say it should be watered a day or two later.  So I looked up the weather forecast, and it said that the chance of rain for the following three days is 50%, 40%, and 30%, so what is the chance that it will rain at some point over the entire 3 day period?  It seems like maybe you could average the three percentages, but I don't think that works, because with tomorrow alone there is a 50% chance, so for the three day period there has to be a minimum of a 50%.  Day two is less than 50%, but it seems like adding another day, even if it is a lower chance, should actually increase the chances over all.  Is there an answer here or not?

Yes, there is. See, the chance that it will rain at least once in the next three days is the same as saying that it will not be the case that it will not rain in the next three days. Assuming that the three are independent, then the chance that there will not be rain in the next three days are 50%, 60%, and 70%. Because these events are independent, the chance of all three days being without rain is the product of these three figures: which is 21%. Finally, the chance that there will be no rain in these three days beign 21% implies that at least one of them will see rain is 79%.

Put another way if there is a 10% chance for 100 days in a row, wouldn't a chance of rain over that many days approach a certainty?

Yes. The chance that it will rain at least once in the next 100 days is 99.9999...% (in fact, the difference between this and certainty is... 1e-100), and will get really really close to certainty (but never achieving it) as you consider more days. But only at the start of those days! Once the first 50 days pass without rain, you might consider watering your lawn.

[Baruch]

Since the weather is a vast chaotic system, independent events really don't apply, particularly after a long period of consecutive time.  It really isn't like flipping a coin, or a multisided D&D dice.

[Atheon]

You're both right, with the first reply doing the math with the assumption that the events are independent. But the reality is that they are not, which complicates the math. You would have to use conditional probabilities (like the odds that it will rain tomorrow given that it rained today). Over three days, if you know all the conditional probabilities, you could draw a probability tree and do the math. The more days involved, the more complex the math becomes.

[Baruch]

You're both right, with the first reply doing the math with the assumption that the events are independent. But the reality is that they are not, which complicates the math. You would have to use conditional probabilities (like the odds that it will rain tomorrow given that it rained today). Over three days, if you know all the conditional probabilities, you could draw a probability tree and do the math. The more days involved, the more complex the math becomes.

Don't let HR know that you know math ... triggers him.  I don't think the OP wanted a lecture in meteorology or chaos mechanics.  I had to think about the problem (doesn't happen much with my posts) ... because I can easily get things backward with ... probability.  It is kind of counter intuitive, and I am better at intuitive.

[Hakurei Reimu]

Don't let HR know that you know math ... triggers him.  I don't think the OP wanted a lecture in meteorology or chaos mechanics.  I had to think about the problem (doesn't happen much with my posts) ... because I can easily get things backward with ... probability.  It is kind of counter intuitive, and I am better at intuitive.

Baruch, I get "triggered" by people NOT knowing math, not by knowing it. I had similar thoughts of not bogging down the explanation, and there's no good way to account for conditionals with the information provided.

[SGOS]

I understand that it can be mathematically solved, as I suspected, but wasn't sure.  So if I decide to spend my time assimilating the process it won't be a waste of time, well at least as a mental exercise.  But I'll still keep my hoses and sprinklers ready.

[trdsf]

Also keep in mind that your probabilities don't stay static -- at first, it's as HR calculated, a 79% chance over the next three days.  But if Day 1 passes and rain doesn't happen, the chance over the next two days is 58% -- (100%-(60%*70%)) -- not the original 79%, because your 50/50 day drops out of the equation.

That's when you should probably start eyeing your hose with intent.

And that last is a sentence that should never be taken out of context.

[Baruch]

I understand that it can be mathematically solved, as I suspected, but wasn't sure.  So if I decide to spend my time assimilating the process it won't be a waste of time, well at least as a mental exercise.  But I'll still keep my hoses and sprinklers ready.

If you have to start using the formula for conditional probabilities ... then you have to pull the correlation coefficients out of your behind ... which are even more sketchy than the original odds.  If you can handle that degree of sketchiness ... you could be an economist ;-)

[SGOS]

If you have to start using the formula for conditional probabilities ... then you have to pull the correlation coefficients out of your behind ... which are even more sketchy than the original odds.  If you can handle that degree of sketchiness ... you could be an economist ;-)

For it what it's worth, it already rained this morning.  So that's already twice the 50% according to my monkeymath.

[Cavebear]

For it what it's worth, it already rained this morning.  So that's already twice the 50% according to my monkeymath.

The odds are the multiplier of the percent for all the days divided by the number of days.  But it is less than that.  The odds of precipitation are for a large area.  YOUR specific odds are much less.  Trust me, I'm a gardener.  I know this practical stuff.

[aitm]

Stupid question....wash car..rain..how hard was that?

[Sorginak]

I've came up with a math problem after applying weed killer to my lawn, and the directions say it should be watered a day or two later.  So I looked up the weather forecast, and it said that the chance of rain for the following three days is 50%, 40%, and 30%, so what is the chance that it will rain at some point over the entire 3 day period?  It seems like maybe you could average the three percentages, but I don't think that works, because with tomorrow alone there is a 50% chance, so for the three day period there has to be a minimum of a 50%.  Day two is less than 50%, but it seems like adding another day, even if it is a lower chance, should actually increase the chances over all.  Is there an answer here or not?

Put another way if there is a 10% chance for 100 days in a row, wouldn't a chance of rain over that many days approach a certainty?

None of that is how the weather works, and conjoining math to it makes it doubly not work.

[Cavebear]

Stupid question....wash car..rain..how hard was that?

Solution: soap car.  Let the rain come...  LOL!