Question 1191712: The probability of rain on first 3 days of august is as follows:
Day probability gain
first day 3/10
2nd day if rain on day 1 is 5/10
if no rain of day 1 is 2/10
3rd day if rain on day 1 and day 2 are 9/10
if rain on day 1 only is 5/10
if rain on day 2 only is 3/10
if no rain on day 1 and day 2 are 5/10
what is the probability of rain on atleast 2 out of 3 days?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The given probabilities, and the probabilities that can be deduced from the given probabilities, are the following:
P(rain on day 1) = 0.3 (given)
P(no rain on day 1) = 1-0.3 = 0.7 (deduced)
P(rain on day 2 given rain on day 1) = 0.5 (given)
P(no rain on day 2 given rain on day 1) = 1-0.5 = 0.5 (deduced)
P(rain on day 2 given no rain on day 1) = 0.2 (given)
P(no rain on day 2 given no rain on day 1) = 1-0.2 = 0.8 (deduced)
P(rain on day 3 given rain on days 1 and 2) = 0.9 (given)
P(no rain on day 3 given rain on days 1 and 2) = 1-0.9 = 0.1 (deduced)
P(rain on day 3 given rain only on day 1) = 0.5 (given)
P(no rain on day 3 given rain only on day 1) = 1-0.5 = 0.5 (deduced)
P(rain on day 3 given rain only on day 2) = 0.3 (given)
P(no rain on day 3 given rain only on day 2) = 1-0.3 = 0.7 (deduced)
P(rain on day 3 given no rain on days 1 or 2) = 0.5 (given)
P(no rain on day 3 given no rain on days 1 or 2) = 1-0.5 = 0.5 (deduced)
We could use those probabilities to answer the question directly, However, it is useful to calculate the probabilities of all possible outcomes to verify that their sum is 1; that will give us confidence we have done the calculations correctly.
Using "Y" (yes) or "N" (no) to indicate whether or not there is rain on each of three days, we have the following probabilities:
P(YYY) = (.3)(.5)(.9) = .135
P(YYN) = (.3)(.5)(.1) = .015
P(YNY) = (.3)(.5)(.5) = .075
P(YNN) = (.3)(.5)(.5) = .075
P(NYY) = (.7)(.2)(.3) = .042
P(NYN) = (.7)(.2)(.7) = .098
P(NNY) = (.7)(.8)(.5) = .280
P(NNN) = (.7)(.8)(.5) = .280
The sum of those probabilities is 1, so it is very likely we have done the calculations correctly.
Then the answer to the problem is the sum of the probabilities that it rains on either 2 or all 3 of the 3 days:
ANSWER: P(either 2 or 3 days of rain) = .135+.015+.075+.042 = .267
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