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?
Please help!!!
$\endgroup$ 14 Answers
$\begingroup$Here is a formula:$P(A\text{ and }B)=P(A)P(B)$ for independent events $A$ and $B$.
$\endgroup$ 4 $\begingroup$P (rain in SAT) = 30% = 3/100 = 0.3 P (rain in SUN) = 45% = 45/100 = 0.45
P ( not rain in both ) = 1 - (P (rain in SAT) * P (rain in SUN)) = 1 - ( 0.3 * 0.45) = 1 - 0.135 = 0.865 P ( rain in both ) = 1 - P ( not rain in both ) = 1 - 0.865 = 0.135
$\endgroup$ $\begingroup$There are four possibilities:
Rain Saturday, Rain Sunday
Rain Saturday, No Rain Sunday
No Rain Saturday, Rain Sunday
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%.
$\endgroup$ 2 $\begingroup$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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