I've been trying to solve the following problem:
After years of observations it has been found out that if it rains on
given day, there is a 60% chance that it will rain on the next day
too. If it is not raining, the chance of rain on the next day is only
25%. The weather forecast for Friday predicts the chance of rain is
75%. What is the probability that at least one day of the weekend will
have no rain?
I calculate the probability that at least one day of the weekend will have no rain as
$$P(\overline{Saturday} \cup \overline{Sunday}) = 1 – P(Saturday \cap Sunday) = 1 – P(Saturday)*P(Sunday)$$
where $P(Saturday)$ and $P(Sunday)$ are the probabilities of raining on Saturday and Sunday respectively.
I compute the probability of raining on Saturday by adding the likelihoods of two exclusive events: raining on Saturday after raining on Friday and raining on Saturday after not raining on Friday.
$$P(Saturday)=P(Saturday \cap Friday)+P(Saturday \cap \overline{Friday}) =P(Saturday|Friday)*P(Friday)+P(Saturday|\overline{Friday})*P(\overline{Friday})=0.6*0.75+0.25*(1-0.75)=0.5125$$
Similarly, I compute the probability of raining on Sunday:
$$P(Sunday)=P(Sunday \cap Saturday)+P(Sunday \cap \overline{Saturday})=P(Sunday|Saturday)*P(Saturday)+P(Sunday|\overline{Saturday})*P(\overline{Saturday})=0.6*0.5125+0.25*(1-0.5125)=0.429375$$
Now I can find the result:
$$P(\overline{Saturday} \cup \overline{Sunday}) = 1 – P(Saturday)*P(Sunday)=1-0.5125*0.429375=0.7799453125$$
Unfortunately, my result doesn't match the expected answer 0.69. Does anyone see a mistake in my calculations?
Best Answer
Denoting Friday, Saturday, Sunday, by the suffixes $1,2,3$ respectively,
P(Rain on both Saturday and Sunday)
$= P(R_1).P(R_2|R_1).P(R_3|R2) +P(R^c_1).P(R_2|R^c_1).P(R_3|R_2)$
$= 0.75*06*0.6 + 0.25*0.25*0.6 = 0.3075$
Thus P(at least one day of weekend has no rain) $=1-0.3075 = 0.6925$