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
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.