On any given day, the probability that it will rain on the island of Sokatoa is estimated to be0.35 . The daily weather is observed on Sokatoa over the course of one month ( 28 consecutivedays).X is the random variable ‘number of days on which it rains’.a Find the probability that it rains on 14 days in total. (2)b Find the largest value of q such that ( ) 0.9P X q≤ < (3)c Given that it rains on 14 days in total, find the probability that it rains on seven days duringthe first two weeks and seven days during the second two weeks.

a) This is a binomial distribution problem. The formula for binomial distribution is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials

In this case, n=28 (the number of days in a month), k=14 (the number of days it rains), and p=0.35 (the probability that it rains on any given day).

So, the probability that it rains on 14 days in total is:

P(X=14) = C(28, 14) * (0.35^14) * ((1-0.35)^(28-14))

b) To find the largest value of q such that P(X<=q) < 0.9, you would need to calculate the cumulative distribution function (CDF) for each possible value of q (from 0 to 28) and find the largest q such that the CDF is less than 0.9. This would typically require a statistical software or calculator.

c) Given that it rains on 14 days in total, the probability that it rains on seven days during the first two weeks and seven days during the second two weeks is:

P(X=7 in first two weeks and X=7 in second two weeks | X=14 in total) = P(X=7 in first two weeks) * P(X=7 in second two weeks)

This is because the number of rainy days in the first two weeks and the second two weeks are independent events. The probability that it rains on seven days in a two-week period can be calculated using the binomial distribution formula as in part a), with n=14 (the number of days in two weeks), k=7 (the number of days it rains), and p=0.35 (the probability that it rains on any given day).