Probability

[duszek]

The probability of raining on Saturday is 50 per cent, and so it is on Sunday.
What is the probability of raining on the week-end ?

75 per cent, according to an expert on TV.

But if we look at the lack of rain we get 50 per cent on Saturday and 50 per cent on Sunday, and on the week-end it is also 75 per cent.

And the two probabilities add up to 150 per cent.

How can it be ?

[mickthinks]

The problem arises from the use of natural language to describe the facts and to translate the mathematical terms.

On the face of it, Xsat = Xsun = 0.5 => Xw/e = 0.75 looks as if it should make sense whatever X stands for, either "rain" or "no rain". It's the whole point of using variables like X. But that is to misunderstand how "no" works.

When X is "rain" then Xw/e occurs when either event occurs, Xsat OR Xsun
When X is "no rain" then Xw/e requires both events, Xsat AND Xsun

Xsat AND Xsun = 0.5 x 0.5 = 0.25
Xsat OR Xsun = 0.5 + 0.5 — (Xsat AND Xsun) = 0.5 + 0.5 — 0.25 = 0.75

...and the two probabilities sum to 1 as you would expect.

[duszek]

Thank you Mick, this makes sense.

However, how about if we considered the simple probability of a dry spam of time on either Saturday OR Sunday.
Because we wish to make pictures of a sky without rain falling out of it.

Then we would have again the probability of 75 percent, wouldn´t we ?

[mickthinks]

Yes, though note that we have also glossed over the matter of what "rain on Saturday" and "rain on Sunday" mean. If we mean "never stops raining during the daylight hours" then, if that has a 50% chance, the chance of a dry spell in which to take a photo of dry weather is 75%.

But of course, we usually mean "some rain", and this would mean there was a good chance of a dry spell on a rainy day. In other words, a 50% chance of rain on Saturday can mean a greater-than-50% chance of a dry spell on Saturday, and hence a greater-than-75% chance of a dry spell during the weekend.

[duszek]

If the rain probability means some rain and we mean by "raining" also a rain lasting 10 minutes in the whole day than the chances of making a picture during a dry spell is 99,999999999 percent, I would guess.

So the chances of not raining (in the sense of a dry spell only) are almost 100 percent.

In this case we could perhaps say:
The chances of raining are 50 percent and of dry spells (almost) 100 percent.

Do we need the "almost" ?

[mickthinks]

Yes, we need the "almost" here—"almost" is almost always needed.

[duszek]

Let´s take something better than weather, a coin flipped has a 50 percent chance of falling heads and 50 percent of falling tails.

What is the chance of heads in two flips ? 50 percent or 75 percent ?
The calculation should be made before the flipping started.