If the probaility of rain on any given day in Chicago during

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Manhattan Prep

If the probability of rain on any given day in Chicago during the summer is 50%, independent of what happens on any other day, what is the probability of having exactly 3 rainy days from July 4th through July 8, inclusive?

A. 1/32
B. 2/25
C. 5/16
D. 8/25
E. 3/4

OA C

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Manhattan Prep

If the probability of rain on any given day in Chicago during the summer is 50%, independent of what happens on any other day, what is the probability of having exactly 3 rainy days from July 4th through July 8, inclusive?

A. 1/32
B. 2/25
C. 5/16
D. 8/25
E. 3/4

OA C

The number of days from July 4 to July 8, inclusive, is 5 days. Thus, we need to determine the probability of having 3 rainy days within a 5-day period. We are given that the probability of a rainy day is ½, and thus the probability of no rain is also 1/2. We need to determine the probability of 3 rainy days within a 5-day period. We can assume the first 3 days are rainy (R) and the last 2 days are not rainy (N). Thus,

P(R-R-R-N-N) = 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32

However, we need to determine in how many ways it can rain 3 out of 5 days. That number will be equivalent to how many ways we can organize the letters R-R-R-N-N.

We use the indistinguishable permutations formula to determine the number of ways to arrange R-R-R-N-N: 5!/(3! x 2!) = 10 ways

Each of these 10 ways has the same probability of occurring. Thus, the total probability is:

10(1/32) = 10/32 = 5/16

Answer: C