If the probability of rain on any given day in Chicago during the summ

Question

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 4 through July 8, inclusive?

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

Show Answer

Official Answer

C

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

KarishmaB · Accepted Answer

Responding to a pm:

There are 5 days from July 4 to July 8 inclusive. You need the probability of 3 rainy and 2 non rainy days.

Method 1:
Probability = Favorable outcomes/Total outcomes

Total outcomes = 2*2*2*2*2 = 32 
On each day, two things are possible - either it is rainy or non rainy. So the 5 days can happen in 2^5 = 32 ways e.g. 
RRNNN, RNNNN, NNRNR, RRRRR etc

Favorable outcomes - 3 R days and 2 N days e.g.
RRRNN, RNRNR, RRNNR etc
 In how many ways can you arrange 3 Rs and 2 Ns? 5!/3!*2! = 10

Probability = 10/32 = 5/16

Method 2:
Probability of rain = 1/2
Probability of no rain = 1/2

Probability of RRRNN = (1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/32
But there are other combinations too such as RRNRN, NNRRR etc. There are 10 such combinations as calculated above (in bold). 
So total probability of 3 Rs and 2 Ns = (1/32) * 10 = 5/16

Thoughtosphere · Answer

From July 4 to July 8, we have total of 5 days. Prob of raining on exactly 3 days =  5C3*(1/2)^3*(1/2)^2

= 5/16

C

EMPOWERgmatRichC · Answer

Hi All,

The most efficient way to answer this question is in understanding how the Combination Formula applies to the math involved. However, even if you don't fully understand that concept, you can still get to the correct answer with a bit of 'brute force.'

To start, you have to understand that since each day has two equally-likely outcomes ('rain' or 'not rain'), then the various arrangements of 5 days of weather = (2)(2)(2)(2)(2) = 32. Thus, the probability of having EXACTLY 3 rain days must be some fraction out of 32...

From this point, we just have to figure out how many options consist of just 3 rain days. We can 'map' those out rather easily...

RRRNN

RRNRN 
RRNNR

RNRRN 
RNRNR 
RNNRR

NRRRN 
NRRNR 
NRNRR 
NNRRR

10 total options out of 32. Reducing that fraction gives us... 10/32 = 5/16

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

Aditi10 · Answer

Hi  VeritasPrepKarishma ,  Bunuel

For someone who is particularly weak at Probability and Combinatorics, where would you recommend practice for ground work?
I have read through the theory but the application for under 700 Questions is what I need to work on.

thanks in anticipation.

KarishmaB · Answer

You can search for relevant tags (Probability, Below 600 and 600 - 700 level) on this forum itself.

Dlc930 · Answer

Can someone explain why the formula is 5C3∗(1/2)^3∗(1/2)^2 instead of 5C3∗(1/2)^3∗(1/2)^ 5

I followed this example on another post on probability:
Example #1
Q.:If the probability of raining on any given day in Atlanta is 40 percent, what is the probability of raining on exactly 2 days in a 7-day period?
Solution: We are not interested in the exact sequence of event and thus apply formula #2:
P=C72∗0.42∗0.65

KarishmaB · Answer

Note that the number of days in the two cases are different. 
In our original question, we are talking about 5 days so probability of rain on 3 days will be (1/2)^3 and of no rain on 2 days will be (1/2)^2.

In the other question you mentioned, we are talking about 7 days so we have (1/2)^2 and (1/2)^5.

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If the probability of rain on any given day in Chicago during the summ

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