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In a particular area, the probability of rain is $0.7$ if it rained the day before. The probability that it does not rain is $0.8$ if it did not rain the day before. It rains on Monday. Find the probability it will be fine on Wednesday. (don't know how to work out)

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Hint: Let $T$ be the event of rain on Tuesday, and $W$ the event of rain on Wednesday. We want $1 - P(W)$. We will first find $P(W)$.

Using conditional probability, $$P(W) = P(W \mid T) P(T) + P(W \mid T^C) (1 - P(T)).$$ Note that $P(W \mid T^C) = 1 - P(W^C \mid T^C)$.

If it rained on Monday, can you fill in the probabilities in this expression?

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you can do this using transition matrix
states are {0 and 1}
0 - does not rain
1 - it rain
P00 = 0.8
P11 = 0.7
P10 = 1- P11 = 0.3
P01 = 1-P00 = 0.2
P = [0.8 0.2 ; 0.3 0.7];

Note that there are two transition, Monday to tuesday and tuesday to wednesday
I = [0 1 ] ;
here I denotes initial condition
I * P^2 = [0.4500 0.5500];
hence
probability that it will rain on wednesday = 0.55
and it does not rain = 0.45

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