Problem Solving

BTGmoderatorLU wrote:Source: Veritas Prep

If there is a 20% chance of rain every day for the next 7 days, what is the probability that it will rain exactly 2 days out of the next 7 days?

$$A. \ \frac{21}{10^7}$$
$$B.\ \frac{2^7}{10^7}$$
$$C.\ \frac{21^7}{10^7}$$
$$D.\ \frac{21\cdot2^7}{10^7}$$
$$E.\ \frac{42\cdot2^7}{10^7}$$
The OA is D

We can assume the scenario is RRNNNNN (i.e., it rains the first 2 days and doesn't rain for the next 5). Therefore, the probability of this scenario is 2/10 x 2/10 x 8/10 x 8/10 x 8/10 x 8/10 x 8/10 = (2/10)^2 x (8/10)^5 = (2^2 x 8^5)/10^7 = (2^2 x 2^15)/10^7 = 2^17/10^7.

However, this is not the only scenario in which it will rain exactly 2 days out of the 7 days. For example, it can rain on the third and fifth days and not on the other 5 days (i.e., NNRNRNN). In fact, there are 7!/(2! x 5!) ways it can rain exactly 2 days out of the 7 days since that is the number of ways we can arrange the 2 R's and 5N's.

Since 7!/(2! x 5!) = (7 x 6)/2 = 21, then the probability that it will rain exactly 2 days out of the 7 days is:

2^17/10^7 x 21 = (21 x 2^17)/10^7

Answer: D