40% + 60% = 100% Right?

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I heard there was a 40% chance of rain this morning and a 60% chance of rain this afternoon. Does this mean there is a 100% chance of rain some time today?

Assuming rain in the morning and rain in the afternoon are completely independent events* (they are not).

We calculate the probability of rain in the morning AND the afternoon to be 0.4 * 0.6 = 0.24 (or 24%).
We calculate the probability of rain in the morning AND no rain in the afternoon to be 0.4 * 0.4 = 0.16 (or 16%).
We calculate the probability of no rain in the morning AND rain in the afternoon to be 0.6 * 0.6 = 0.36 (or 36%)

Adding up all the probabilities of some combination of rain and not rain we get 24% + 16% + 36% = 76% chance of rain today.

Alternatively, it tends to be more commonly calculated as the (1 - probability of no rain**), which is 1.0 - 0.6 * 0.4 = 1.0 – 0.24 = 0.76 = 76% chance of rain.

* Independent events are events whose outcome (result) has no relationship to previous outcomes (results). For example, tossing a coin: whether the coin comes up heads or tails does not depend on what the coin came up before.

** the probability of no rain in the morning is 60% (or 0.6) and the probability of no rain in the afternoon is 40% (or 0.4) in case you were wondering where those number came from.

Bonus: 20+ years ago I discovered that weather forecasters calculate the probability of rain based on historical meteorological records. If they say there is a 40% chance of rain, what they are really saying is, "Based on our records of similar weather conditions, it rained 40% of the time". As far as I know, there is no magic formula they use, just a big database look up.

Image is nabbed from here.

Comments

Ulysses said…
I often think the forecast should be given as both the predicted possibility and the expected coverage. Is a 40% chance of rain an 100% chance of rain over 40% of the local area, or a 40% chance of rain over the entire covered area?
For sure, the garden is sure suffering...
July 12, 2007 8:05 PM
Richard said…
ulysses: I doubt very much they would go for that because it treads too far into the domain of geography. My observation is that people like to keep to their own little fiefdoms.

I think your garden needs more drought tolerant crops. My front lawn in Ottawa is still pretty green (and I don't need to water) because it is heavy with clover (which resists dry conditions better than grass).
July 13, 2007 10:31 AM
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