OA IS C!

Answer

The probability that we are being asked to calculate here is comprised of multiple scenarios. Put differently, there are a number of ways for it to rain at least 5 days in a row. To solve for the overall probability, we must find the probability for each one of these independent scenarios and add them together.

Letâ€™s set: R = rainy day

S = sunny day

There are 3 different types of scenarios that we need to consider: (1) 5Râ€™s 2Sâ€™s (2) 6Râ€™s 1S and (3) 7Sâ€™s. Of course we must remember that within each scenario it must rain for at least 5 consecutive days.

TYPE 1: There are 3 ways for it to rain exactly 5 out of 7 days and for the 5 days to be consecutive: RRRRRSS, SSRRRRR, SRRRRRS (two Sâ€™s at the end, the beginning, or one on each end).

TYPE 2: There are 4 ways for it to rain exactly 6 out of 7 days and for at least five of the rainy days to be consecutive: RRRRRRS, SRRRRRR, RSRRRRR, RRRRRSR (S in position 1, 2, 6 or 7).

TYPE 3: There is one way for it to rain exactly 7 out of 7 days: RRRRRRR.

Now that we have counted and identified the different scenarios, we must find the probability of each scenario (or at least of each type of scenario).

TYPE 1: If the probability of R is Â¾ and the probability of S is Â¼, the probability of 5Râ€™s and 2Sâ€™s (it doesnâ€™t matter what order) is (3/4)^5(1/4)^2. There are three such scenarios so this contributes 3(3/4)^5(1/4)^2 to the final probability.

TYPE 2: If the probability of R is Â¾ and the probability of S is Â¼, the probability of 6Râ€™s and 1S (it doesnâ€™t matter what order) is (3/4)^6(1/4)^1. There are four such scenarios so this contributes 4(3/4)^6(1/4) to the final probability.

TYPE 3: If the probability of R is Â¾ and the probability of S is Â¼, the probability of 7Râ€™s is (3/4)^7. There is only one such scenario so this contributes (3/4)^7 to the final probability.

The final probability is 3(3/4)^5(1/4)^2 + 4(3/4)^6(1/4) + (3/4)^7.

The correct answer is C.

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"Wow! Brazil is big." â€”George W. Bush, after being shown a map of Brazil by Brazilian president Luiz Inacio Lula da Silva, Brasilia, Brazil, Nov. 6, 2005

http://www.nytimes.com/2005/11/21/international/asia/21prexy.html