# If there is a 30% chance of rain for both Saturday and Sunday, what is the probability there would be rain on both days?

Can someone give me a formula for this question?

Wanda said there was a 30% chance of rain for both Saturday and Sunday. It rained on both days.

If her calculations were correct and there was a 30% chance of rain each day, what is the probability there would be rain on both days?

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Here is a formula: $P(A\text{ and }B)=P(A)P(B)$ for independent events $A$ and $B$.

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thank you, but what does that mean? – raven May 22 '13 at 22:48
@ZevChonoles What is your basis for assuming the events are independent? – Dale M May 23 '13 at 6:14
@DaleM: That the problem would not be answerable without this assumption. Should whoever wrote the question have said the events were independent explicitly? Yes. Should they have used an example from the real world where the events were more obviously independent than the weather on consecutive days? Yes. But that doesn't mean we can't have a good idea of what they meant. – Zev Chonoles May 23 '13 at 6:16
@ZevChonoles I can see what they meant, too. I just think it is bad idea to make assumptions that are self-evidently not true. – Dale M May 23 '13 at 6:18

There are four possibilities:

1. Rain Saturday, Rain Sunday

2. Rain Saturday, No Rain Sunday

3. No Rain Saturday, Rain Sunday

4. No Rain Saturday, No Rain Sunday

You can calculate the probability of each of these four cases using @Zev's formula; note that their sum will be 100%.

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What is your basis for assuming the events are independent? – Dale M May 23 '13 at 6:14
@DaleM, because it is evident from the problem that this is a junior-high level problem, and "dependent events" are over the horizon. – vadim123 May 23 '13 at 13:21

This is an appallingly constructed problem and is likely to lead to confusion and error for anyone trying to learn from it.

For two events $A$ and $B$ with probabilities $P(A)$ and $P(B)$, the probability of them both occurring - technically the intersection of the events $A \cap B$ is

$$P(A\cap B)=P(A)P(B|A)=P(B)P(A|B)$$

For the particular example, this says that the probability of it raining on both days is the probability of it raining on Saturday times the probability that it rains on Sunday given that it has rained on Saturday (or vice-versa).

Now, for independent events, that is that the fact that one has (or has not) occurred does not affect the other, $P(A|B)=P(A)$. However, the chance of rain on two consecutive days in the same location are not independent, not even close.

What the question wants you to do is multiply $0.3$ by $0.3$, but this is soooo wrong that it is frightening. People have been sent to goal (or jail if you prefer) for decades because so called experts cannot tell the difference between independent and correlated events.

Please note that the relationship between these events does not necessarily have to be causal. Rain is an effect of atmospheric conditions - these generally last longer than 24 hours so if it rains today it is more likely to rain tomorrow - this doesn't mean that today's rain caused tomorrows - they are simply different effects of another cause.

The correct answer to this problem is:

$$P(\text{rain on both days})=P(\text{rain Saturday})P(\text{rain Sunday}|\text{rain Saturday}) = 0.3 \times ?$$

You are not given enough information to go further!

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