in a beach side suburb it rains on 20% of days and is windy on 30% of days. if rain and wind are independent, on any particular day find the probability that:
(a) it rains and is windy (b) it does not rain and is not windy (c) it rains or it windy (d) it does not rain or is not windy
Start with a table:
R=rain (20%), ~R=no rain (80%)
W=windy (30%), ~W=not windy (70%)
If rain and wind are independent, the combined occurrence is the product of the respective probabilities.
Example:
(a)P(R∧W)=.2*.3=0.06
(b)P(~R∧~W=0.8*0.7=0.56
(c)...
Will leave (c) and (d) for you as exercise.
(a) .20 * .30 = .06
(b) (1-.20) * .30 = .24
(c) .20 + .30 - .20*.30 = .44
(d) (1-.20) + (1-.30) - (1-.20)(1-.30) = .94
Note that (d) = 1-(a)
To find the probability in each scenario, we can simply apply the rules of probability. Let's calculate each probability step by step:
(a) To find the probability that it rains and is windy, given that rain and wind are independent events, we can multiply their respective probabilities. So, P(it rains and is windy) = P(rain) * P(wind).
Given:
P(rain) = 0.20 (since it rains on 20% of days)
P(wind) = 0.30 (since it is windy on 30% of days)
Therefore, P(it rains and is windy) = 0.20 * 0.30 = 0.06 or 6%.
(b) To find the probability that it does not rain and is not windy, we can find the complement of the events "rain" and "wind."
P(it does not rain) = 1 - P(rain) = 1 - 0.20 = 0.80 (since it does not rain on 80% of days)
P(it is not windy) = 1 - P(wind) = 1 - 0.30 = 0.70 (since it is not windy on 70% of days)
Now, to find the probability that it does not rain and is not windy, we multiply these probabilities:
P(it does not rain and is not windy) = P(it does not rain) * P(it is not windy) = 0.80 * 0.70 = 0.56 or 56%.
(c) To find the probability that it rains or is windy, we can use the addition rule of probability. However, we need to consider that rain and wind can occur simultaneously, so we should subtract the probability of rain and wind occurring at the same time to avoid double-counting.
P(it rains or is windy) = P(rain) + P(wind) - P(it rains and is windy)
= 0.20 + 0.30 - 0.06
= 0.44 or 44%.
(d) To find the probability that it does not rain or is not windy, we can again use the complement rule.
P(it does not rain or is not windy) = 1 - P(it rains and is windy)
= 1 - 0.06
= 0.94 or 94%.
So, to summarize:
(a) P(it rains and is windy) = 0.06 or 6%
(b) P(it does not rain and is not windy) = 0.56 or 56%
(c) P(it rains or is windy) = 0.44 or 44%
(d) P(it does not rain or is not windy) = 0.94 or 94%