In: Statistics and Probability
suppose there is a 32% chance of rain for each of the next two days, find the following probabilities.
a. it doesn't rain tomorrow?
b it rains tomorrow and the next day? consider whether independent or dependent.
c. it rains tomorrow or the next day? consider whether mutually exclusive or not.
d. it rains at least once over the next two days?
Given that the probability of rain for each of the next two days is 32% or 0.32 .
a. Now, the probability for a rain tomorrow is 0.32, thus the probability that it doesn't rain tomorrow is 1-0.32= 0.68
b. Now, a rain tomorrow and the next day are not dependent events since happening on rain the next day does not depend upon happening of rain tomorrow.
Hence, the probability that it rains tomorrow and the next day= P(it rains tomorrow)* P(it rains the next day)= 0.32*0.32= 0.1024
c. Again, a rain tomorrow or the next day are not mutually exclusive since occurrence of one does not cancel the occurrence of the other.
Hence, the probability that it rains tomorrow or the next day= P(it rains tomorrow)+ P(it rains the next day)- P(it rains tomorrow and the next day)
= 0.32+ 0.32- 0.1024= 0.5376
d. Since the probabilities of rain on both days are equal or constant, we can model this situation as a binomial experiment, considering the rain on a particular day as success and no rain as failure. Let X denote the event that it rains on a particular day, then X is a binomial random variable with success probability p= 0.32. Here no. of trials n= 2, since we are considering the next two days.
The PMF of X is given as
P(X=x)= 2Cx* (0.32)^x* (0.68)^(2-x) for x=0,1,2
Now, we need the probability that it rains at least once over the next two days or P(X>=1)= 1- P(X<1)= 1- P(X=0)= 1- { 2C0* (0.32)^0* (0.68)^2}
= 1- 0.68*0.68= 1- 0.4624 = 0.5376