Observation of weather conditions in a certain region of the country shows that
ID: 3124495 • Letter: O
Question
Observation of weather conditions in a certain region of the country shows that it rains 40 % of the time and shines 60 % of the time. A barometer manufacturer has compared the coincidence of his barometer predictions "rain" and "shine" with the actual Weather States: Rain and Shine and discovered that the readings of his instruments are correct 90 % of the time when it Rains and 80 % of the time when it shines What is the probability it will rain when his instrument says "rain" (what is it if the instrument predicts "shine") ? What is the probability it will shine when his instrument says "shine" (what is it when his instrument predicts "rain") ? What is the probability that the barometer reads correctly?Explanation / Answer
Let
R, S = actual rain, shine
r, s = instrument reads rain, shine
a)
As
P(r) = P(R) P(r|R) + P(S) P(r|S) = 0.40*0.90 + 0.60*(1-0.80) = 0.48
Hence,
P(R|r) = P(R) P(r|R) / P(r)
= 0.40*0.90/0.48
= 0.75 [ANSWER]
***********************
b)
P(S|s) = P(S) P(s|S)/P(s)
= 0.60*0.80/(1-0.48)
= 0.923076923 [ANSWER]
********************
c)
P[(R n r) U (S n s)] = P(R) P(r|R) + P(S) P(s|S)
= 0.40*0.90 + 0.60*0.80
= 0.84 [ANSWER]
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