Question

Suppose that whether it rains tomorrow depends on past weather conditions only through the last 2 days. Specifically, suppose that if it has rained yesterday and today, then it will rain tomorrow with probability .8; if it rained yesterday but not today, then it will rain tomorrow with probability .3; if it rained today but not yesterday, then it will rain tomorrow with probability .4; and if it has not rained either yesterday or today, then it will rain tomorrow with probability .2. Over the long term, what proportion of days does it rain

Answer 1

The answer is "0.425".

Explanation:

The T be a requirement today that such rains.

The Y be a situation yesterday that now it rained.

The O is a condition for tomorrow to rain.

`P(O| (Y\ AND \ T)) = 0.8\\\\P(O|(Y \ AND \ NOT\ (T)) =0.3\\\\P( O|(NOT \ (Y)\ AND\ T)= 0.4\\\\P( O| NOT\ (Y) \ OR \ NOT \ (T)) =0.2`

There Arena and morrow overview To Aspects Which will rain:

`Y\ AND \ T, Y \ AND \ NOT \ (T),\ NOT\ (Y) \ AND\ T, \ NOT(Y)\ OR \ NOT(T)`

Each has the option of
`(1)/(4)` INDEPENDENTLY happening.

Its possibility of tomorrow's rain is:

`P(O) = P(O | Y\ AND\ T) P(Y\ AND\ T) + P(O|(Y \ AND \ NOT\ (T))`

`P((Y\ AND \ NOT\ (T)) + P(O|(\ NOT\ (Y) \ AND \ T))P(\ NOT\ (Y) \ AND\ T) \\\\+ P( O| \ NOT\ (Y)\ OR\ NOT\ (T)) P(\ NOT\ (Y)\ OR\ NOT\ (T))\\\\`

`=0.8* (1)/(4) +0.3* (1)/(4)+0.4* (1)/(4)+0.2* (1)/(4)\\\\ =(1)/(4)(0.8+0.3+0.4+0.2)\\\\= (1.7)/(4)\\\\= 0.425`