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Rain check

by AIM GMAT » Sat Mar 19, 2011 11:38 pm
If the probability of rain on any given day is 50%, what is the probability that it
will rain on at least 3 days in a row during a 5 day period?

(A) 3/32

(B) 1/4

(C) 9/32

(D)5/16

(E) 1/2

OA : later with explanation .
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by GMATGuruNY » Sun Mar 20, 2011 12:21 am
AIM GMAT wrote:If the probability of rain on any given day is 50%, what is the probability that it
will rain on at least 3 days in a row during a 5 day period?

(A) 3/32

(B) 1/4

(C) 9/32

(D)5/16

(E) 1/2

OA : later with explanation .
P = (good outcomes)/(total possible outcomes)

Let R = rain, N = no rain.

Total possible outcomes:
Each day could be R or N, giving us 2 possible outcomes for each day.
Over 5 days, total possible outcomes = 2*2*2*2*2 = 32.

Good outcomes:
Exactly 3 days of rain in a row:
RRRNN
RRRNR
NRRRN
NNRRR
RNRRR
5 good outcomes.

Exactly 4 days of rain in a row:
RRRRN
NRRRR
2 good outcomes.

Exactly 5 days of rain in a row:
RRRRR
1 good outcome.

Good outcomes = 5+2+1 = 8.

Thus, P(at least 3 days of rain in a row) = 8/32 = 1/4.

The correct answer is B.
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by vaflaly » Tue Mar 22, 2011 3:56 pm
GMATGuruNY wrote:
AIM GMAT wrote:If the probability of rain on any given day is 50%, what is the probability that it
will rain on at least 3 days in a row during a 5 day period?

(A) 3/32

(B) 1/4

(C) 9/32

(D)5/16

(E) 1/2

OA : later with explanation .
P = (good outcomes)/(total possible outcomes)

Let R = rain, N = no rain.

Total possible outcomes:
Each day could be R or N, giving us 2 possible outcomes for each day.
Over 5 days, total possible outcomes = 2*2*2*2*2 = 32.

Good outcomes:
Exactly 3 days of rain in a row:
RRRNN
RRRNR
NRRRN
NNRRR
RNRRR
5 good outcomes.

Exactly 4 days of rain in a row:
RRRRN
NRRRR
2 good outcomes.

Exactly 5 days of rain in a row:
RRRRR
1 good outcome.

Good outcomes = 5+2+1 = 8.

Thus, P(at least 3 days of rain in a row) = 8/32 = 1/4.

The correct answer is B.
I think you did a mistatke in counting the number of outcomes for 3 daus and 4 days. Actually, for exactly 3 days ----> there are 10 outcomes( 5!/(3!*2!)); for exactly 4 days, there are 5 outcomes ( 5!/(4!*1!)). The total is 10+5+1=16

Thus, P = 16/32 = 1/2. the answer is E.

Using pascal triangle amounts to the same result ( 1/2)

Is that correct??
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by gaurav2011 » Tue Mar 22, 2011 6:17 pm
My answer is 9/32. Please see below my way of reasoing. The key to note is raining at least 3 days "In a row"

1) At least 3 days in a row = In this case, we can consider three continous days raining as one unit and total distinct units wil be 3. We are arrange 3 units in 6 ways hence probability = 6/32.

2) At least 4 days in a row = In this case, we can consider four continous days raining as one unit and total distinct units wil be 2. We are arrange 2 units in 2 ways hence probability = 2/32.

3) At least 5 days in a row = In this case, we can consider five continous days raining as one unit and total distinct units wil be 1. We are arrange 1 unit in 1 way hence probability = 1/32.

Becuase all three cases above are mutually exclusive, the sum of probability rule will apply hence total probability of raning at least 3 days in a row will be 6/32 + 2/32 + 1/32 = 9/ 32 hence ANSWER = C
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by GMATGuruNY » Tue Mar 22, 2011 6:47 pm
vaflaly wrote:
GMATGuruNY wrote:
AIM GMAT wrote:If the probability of rain on any given day is 50%, what is the probability that it
will rain on at least 3 days in a row during a 5 day period?

(A) 3/32

(B) 1/4

(C) 9/32

(D)5/16

(E) 1/2

OA : later with explanation .
P = (good outcomes)/(total possible outcomes)

Let R = rain, N = no rain.

Total possible outcomes:
Each day could be R or N, giving us 2 possible outcomes for each day.
Over 5 days, total possible outcomes = 2*2*2*2*2 = 32.

Good outcomes:
Exactly 3 days of rain in a row:
RRRNN
RRRNR
NRRRN
NNRRR
RNRRR
5 good outcomes.

Exactly 4 days of rain in a row:
RRRRN
NRRRR
2 good outcomes.

Exactly 5 days of rain in a row:
RRRRR
1 good outcome.

Good outcomes = 5+2+1 = 8.

Thus, P(at least 3 days of rain in a row) = 8/32 = 1/4.

The correct answer is B.
I think you did a mistatke in counting the number of outcomes for 3 daus and 4 days. Actually, for exactly 3 days ----> there are 10 outcomes( 5!/(3!*2!)); for exactly 4 days, there are 5 outcomes ( 5!/(4!*1!)). The total is 10+5+1=16

Thus, P = 16/32 = 1/2. the answer is E.

Using pascal triangle amounts to the same result ( 1/2)

Is that correct??
Your approach would be mathematically sound if the question asked for the probability that it will rain on at least 3 days.
But the problem above asks for the probability that it will rain at least 3 days IN A ROW.
If you revisit my solution above, all of the different ways are listed, so there is no room for error.
Last edited by GMATGuruNY on Sun Dec 23, 2012 7:26 pm, edited 1 time in total.
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by vaflaly » Tue Mar 22, 2011 10:41 pm
GMATGuruNY wrote:
vaflaly wrote:
GMATGuruNY wrote:
AIM GMAT wrote:If the probability of rain on any given day is 50%, what is the probability that it
will rain on at least 3 days in a row during a 5 day period?

(A) 3/32

(B) 1/4

(C) 9/32

(D)5/16

(E) 1/2

OA : later with explanation .
P = (good outcomes)/(total possible outcomes)

Let R = rain, N = no rain.

Total possible outcomes:
Each day could be R or N, giving us 2 possible outcomes for each day.
Over 5 days, total possible outcomes = 2*2*2*2*2 = 32.

Good outcomes:
Exactly 3 days of rain in a row:
RRRNN
RRRNR
NRRRN
NNRRR
RNRRR
5 good outcomes.

Exactly 4 days of rain in a row:
RRRRN
NRRRR
2 good outcomes.

Exactly 5 days of rain in a row:
RRRRR
1 good outcome.

Good outcomes = 5+2+1 = 8.

Thus, P(at least 3 days of rain in a row) = 8/32 = 1/4.

The correct answer is B.
I think you did a mistatke in counting the number of outcomes for 3 daus and 4 days. Actually, for exactly 3 days ----> there are 10 outcomes( 5!/(3!*2!)); for exactly 4 days, there are 5 outcomes ( 5!/(4!*1!)). The total is 10+5+1=16

Thus, P = 16/32 = 1/2. the answer is E.

Using pascal triangle amounts to the same result ( 1/2)

Is that correct??
Your approach would be mathematically sound if the question asked for the probability that it will rain on at least 3 days. Since the question asks for the probability that will rain on at least 3 days in a row, it's safer to write out the different ways in which a good outcome could be achieved. If you revisit my solution above, all the different ways are listed, so there is no room for error.

I understand my mystake now from your explaination. the key word is "in the row".
thank you
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by Amrabdelnaby » Fri Nov 27, 2015 10:33 am
Hi Guru,

I see that you solved it by listing... is there another faster way to solve it using probability rules or combinations?

please advise
GMATGuruNY wrote:
vaflaly wrote:
GMATGuruNY wrote:
AIM GMAT wrote:If the probability of rain on any given day is 50%, what is the probability that it
will rain on at least 3 days in a row during a 5 day period?

(A) 3/32

(B) 1/4

(C) 9/32

(D)5/16

(E) 1/2

OA : later with explanation .
P = (good outcomes)/(total possible outcomes)

Let R = rain, N = no rain.

Total possible outcomes:
Each day could be R or N, giving us 2 possible outcomes for each day.
Over 5 days, total possible outcomes = 2*2*2*2*2 = 32.

Good outcomes:
Exactly 3 days of rain in a row:
RRRNN
RRRNR
NRRRN
NNRRR
RNRRR
5 good outcomes.

Exactly 4 days of rain in a row:
RRRRN
NRRRR
2 good outcomes.

Exactly 5 days of rain in a row:
RRRRR
1 good outcome.

Good outcomes = 5+2+1 = 8.

Thus, P(at least 3 days of rain in a row) = 8/32 = 1/4.

The correct answer is B.
I think you did a mistatke in counting the number of outcomes for 3 daus and 4 days. Actually, for exactly 3 days ----> there are 10 outcomes( 5!/(3!*2!)); for exactly 4 days, there are 5 outcomes ( 5!/(4!*1!)). The total is 10+5+1=16

Thus, P = 16/32 = 1/2. the answer is E.

Using pascal triangle amounts to the same result ( 1/2)

Is that correct??
Your approach would be mathematically sound if the question asked for the probability that it will rain on at least 3 days.
But the problem above asks for the probability that it will rain at least 3 days IN A ROW.
If you revisit my solution above, all of the different ways are listed, so there is no room for error.
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by GMATGuruNY » Fri Nov 27, 2015 2:18 pm
Amrabdelnaby wrote:Hi Guru, I see that you solved it by listing... is there another faster way to solve it using probability rules or combinations?

please advise
For most test-takers, the solution above is likely to be the easiest and fastest approach.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
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