Help an old man with problem from Bertsekas
Hello, I am trying to self-study probability as an old old man. I am using the book by Bertsekas and Tsitsiklis, and following the MIT course https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ .
It's quite difficult, and I am stuck on this problem here, where I don't know why a particular way of solving the problem doesn't work. The problem is problem 1 of problem Set 2 of the above course.
I would appreciate any help.
1.
Most mornings, Victor checks the weather report before deciding whether to carry an umbrella. If the forecast is “rain,” the probability of actually having rain that day is 80%. On the other hand, if the forecast is “no rain,” the probability of it actually raining is equal to 10%. During fall and winter the forecast is “rain” 70% of the time and during summer and spring it is 20%.
(a)
Oneday, Victor missed theforecast andit rained. Whatistheprobability thattheforecast was “rain” ifitwasduring thewinter?Whatistheprobability thattheforecastwas “rain” if it was during the summer?
So I can solve it and get the answer in the solutions, using the method they also use in the solutions:
However, I'd like to know why my solution which does not fix the season first is wrong. Here it is:
S=Summer, W=Winter; FR=ForecastRain, ~FR=Forecase no rain; R=Rain, ~R=not rain
P(FR | R ∩ W) = P(FR ∩ R ∩ W ) / P(R ∩ W ) ... definition of conditional probability
P(FR ∩ R ∩ W ) = P(W)P(FR|W)P(R|FR**∩** W) ... Multiplication rule
P(R ∩ W) = P(R ∩ W|FR)P(FR) + P(R ∩ W|~FR)P(~FR) ... Total probability rule
P(FR) = P(FR|W)P(W) + P(FR|S)P(S)
P(~FR) = P(~FR|W)P(W) + P(~FR|S)P(S) = 1 - P(FR)
Then taking into account wh at we are giving, and assigning P(S) = P(W) = 1/2:
P(R|FR) = 0.8; P(R| ~FR) = 0.1
P(FR|W) = 0.7; P(FR|S) = 0.2
P(FR ∩ R ∩ W ) = P(W)P(FR|W)P(R|FR**∩** W) = (1/2)(7/10)(8/10) = 56/200
P(FR) = P(FR|W)P(W) + P(FR|S)P(S) = (7/10)(1/2) + (2/10)(1/2) = 9/20
P(~FR) = 11/20
P(R ∩ W) = P(R ∩ W|FR)P(FR) + P(R ∩ W|~FR)P(~FR) = (8/10)(9/20) + (1/10)(11/20) = (72+11)/200 = 83/200
=> P(FR | R ∩ W) = P(FR ∩ R ∩ W ) / P(R ∩ W ) = 56/83
And as you can see from the above solution this doesn't work.
So What am I getting wrong here? I would appreciate any help, because I think this could reveal some fundamental misunderstanding I have about how the things work.
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I figured it out:
P(R∩ W) = P(R ∩ W|FR)P(FR) + P(R ∩ W|~FR)P(~FR)
= P(R ∩ W∩ FR) + P(R∩ W∩ ~FR)
= 56/200 + P(R∩ W∩ ~FR)
P(R∩ W∩ ~FR) = P(W)P(~FR|W)P(R|W∩ ~FR) ... again multiplication rule
= 3/200
P(R∩ W) = (56+3)/200 = 59/200
=> P(FR|R ∩ W) = P(FR ∩ R ∩ W ) / P(R ∩ W ) = 56/59
The mistake was where I said P(R ∩ W|FR) = 8/10 and P(R ∩ W|~FR), in the model I had set up, the forecast will affect the probability of whether or not it is winter or summer.
This also demonstrates that everything posted about why this method wasn't possible is not correct.