# Probability of rain given various conditions

The probability of rain given rain the proceeding day is $.6$ The probability of rain given no rain the proceeding day is $.25$

The probability of rain for Friday is $.75$

What is the probability there will be no rain on at least one day of the weekend (Sat, Sun)?

My original thought was that one would need to run through all permutations (Fri, Sat, Sun) of raining and not raining then sum the probabilities together. But this seems like a really long problem. This was a question on a Udacity video , but aside from walking through all combinations I'm stumped on how to accomplish this.

So, if i used the notation $$P(r|r) = .6 , P(r|\sim r) = .25$$ $$P(\sim r|r) = .4 , P(\sim r|\sim r) = .75$$ $$P(Fri) = .75$$

Then you would use these to figure out,

$$\begin{gather} P(Fri) \times P(\sim Sat) + P(\sim Fri)\times P(\sim Sat) + P(Fri)\times P(Sat)\times P(\sim Sun) \\ + P(Fri) \times P(\sim Sat) \times P(\sim Sun)\end{gather}$$

...

Not sure if this is correct or if there is a better way

• You use the word proceeding, do you mean to use the word preceding instead? Is it the day before that influences the probability or is the day after? – JMoravitz Feb 25 '17 at 20:10
• Hint: It is easier to find the probability that it will rain on both Saturday and on Sunday, and subtract this number from $1$. I will leave it to you to figure out why such a calculation makes sense, actually do the necessary computations, and then write up everything neatly as an answer to the question that you post yourself to your own question. I also recommend that you edit your question to use MathJax and use MathJax tp impose your answer too. – Dilip Sarwate Feb 25 '17 at 20:10

## 1 Answer

Let $R_5,R_6,R_7$ be the events that it rains on Friday, Saturday, and Sunday, respectively. Then the probability you want is \begin{split} 1-P(R_6,R_7) &= 1-P(R_6,R_7 \vert R_5)P(R_5)-P(R_6,R_7 \vert \sim R_5)P( \sim R_5)\\ &=1-P(R_7 \vert R_6)P(R_6 \vert R_5)P(R_5)-P(R_7 \vert R_6) P(R_6 \vert \sim R_5)P(\sim R_5) \end{split} which follows by a conditioning argument.