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Intern  Joined: 25 Sep 2007
Posts: 17
If the probability of rain on any given day in City X is 50 percent  [#permalink]

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31 00:00

Difficulty:   15% (low)

Question Stats: 75% (01:34) correct 25% (01:50) wrong based on 475 sessions

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If the probability of rain on any given day in City X is 50 percent, what is the probability that it rains on exactly 3 days in a 5-day period?

(A) 8/125
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4

Originally posted by JuliaS on 04 Jan 2008, 16:01.
Last edited by Bunuel on 05 Feb 2019, 05:08, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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7
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flokki wrote:
If the probability of rain on any given day in City X is 50 percent, what is the probability that it rains on exactly 3 days in a 5-day period?

(A) 8/125
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4

Could you tell me why it is a combination and not a permutation?

N = 3C5 x 1/8 x 1/4

why do i have to divide by 2! ?

The probability of rain each day is 1/2 and the probability of no rain is also 1/2. $$C^3_5=10$$ represent ways to choose on which 3 days out of 5 there will be a rain, so $$P=C^3_5*(\frac{1}{2})^3*(\frac{1}{2})^2=\frac{10}{32}=\frac{5}{16}$$.

Or think about it this way: we want the probability of the following event: RRRNN, where R represent rain day and N represents no-rain day. Now, each R and each N have individual probability of 1/2, so $$(\frac{1}{2})^5$$.

But the case of RRRNN can occur in many ways: RRRNN, RRNRRN, RNRRN, NRRRN, ... basically it will be equla to # of arrangements (permutations) of 5 letters RRRNN out of which there are 3 identical R's and 2 identical N's. That # of arrangements is $$\frac{5!}{3!2!}$$, (notice that it's the same as $$C^3_5$$). So, finally $$P=\frac{5!}{3!2!}*(\frac{1}{2})^5=\frac{5}{16}$$.

For more on this topic check Combinations and Probability chapters of Math Book:
math-combinatorics-87345.html
math-probability-87244.html

Also check similar questions to practice:
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on-saturday-morning-malachi-will-begin-a-camping-vacation-100297.html
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combination-ps-55071.html
the-probability-that-a-family-with-6-children-has-exactly-88945.html?

Hope it helps.
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Manager  Joined: 27 Oct 2008
Posts: 130
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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5
3
If the probability of rain on any given day in City X is 50 percent, what is the probability that it rains on exactly 3 days in a 5-day period?

(A) 8/125
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4

Soln:
Let Y represent raining on a day and N represent Not raining on a day
So for it to rain 3 out of 5 days we have YYYNN.
Since the order in which it rains in 5 days matters hence we find the total number of possibilities
But since Y has a happened 3 times and N has happened 2 times we cannot distinguish between them and hence divide by the repetitions . Thus we need to divide by (3! * 2!)
=> 5!/(2! * 3!)
=> 10 ways

Now since on each day it can rain or it cannot rain. Thus total outcomes is
=> 2 * 2 * 2 * 2 * 2
=> 32

So probability that it rains on exactly 3 days is
= 10/32
= 5/16

Ans is C
##### General Discussion
Senior Manager  Joined: 08 Jun 2007
Posts: 430
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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JuliaS wrote:
If the probability of rain on any given day in City X is 50 percent, what is the probability that it rains on exactly 3 days in a 5-day period?

(A) 8/125
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4

Thanks a lot!

welcome to the forum

total possibilties 2^5 = 32
rain combinations 5c3 = 10
10/32 = 5/16
Director  Joined: 01 May 2007
Posts: 644
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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Why is total possibilties 2^5 = 32?
Intern  Joined: 30 Dec 2007
Posts: 6
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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vscid wrote:
yeah how did u arrive at 2^5.

Let me try to elaborate
Each day, we have 2 possibilities, rain or not rain, therefore the total possibility of 5 days is 2^5=32
Then, we pick 3 rainy days randomly from 5 days, 5C3 = 10
10/32=5/16
Manager  Joined: 22 Jul 2009
Posts: 147
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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6
Binomial distribution formula: C(n,k) * p^k * (1-p)^(n-k)

Given that the probability of Rain happening is p (=1/2) and not happening is 1-p (=1-1/2=1/2),
=> Probability of Rain happening k times (=3) in n repeated tests (=5)
= C(5,3) * (1/2)^3 * (1-1/2)^(5-3)
= C(5,3) * (1/2)^5
= 10/32
= 5/16
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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Hi,

I'm confused by the answers, only one answer actually has the 50% probability factored in. How come all the other answers just seemed to mention how many ways to choose 3 days in 5 days?

Seems like 50% is a special case, because when you do 5c3, the 3 days could be 3 days of rain or even 3 days of no rain. They all end up being the same probability because of the 50%.

how would this be solved with 20%? 80%?
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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pinchharmonic wrote:
Hi,

I'm confused by the answers, only one answer actually has the 50% probability factored in. How come all the other answers just seemed to mention how many ways to choose 3 days in 5 days?

Seems like 50% is a special case, because when you do 5c3, the 3 days could be 3 days of rain or even 3 days of no rain. They all end up being the same probability because of the 50%.

how would this be solved with 20%? 80%?

Eg: For the 20% possibility of rain it will be (1/5)^3 * (4/5)^2* (5C3)
Intern  Joined: 01 Dec 2011
Posts: 2
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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Could you tell me why it is a combination and not a permutation?

N = 3C5 x 1/8 x 1/4

why do i have to divide by 2! ?
Intern  Joined: 01 Dec 2011
Posts: 2
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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I just started with permutations and probability.

I still dont understand why we use the combination formula and not the permutation one.

Is it because we do not differentiate between one rainy day and the other? are they all the same for us?

Are most Gmat questions at this level?
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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flokki wrote:

I just started with permutations and probability.

I still dont understand why we use the combination formula and not the permutation one.

Is it because we do not differentiate between one rainy day and the other? are they all the same for us?

Are most Gmat questions at this level?

Yes, we do not differentiate between RRR and NN: for example RRRNN means that it was raining on the first three days and we have no reason to differentiate between them. Also notice that most GMAT combination/probability questions are fairly straightforward and can be solved in several ways. This problem is also dealing with a simple concept explained in the links I sited above, so I'd recommend to brush your fundamentals on combinations before you move to the problems.
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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Binomial formula b(x; n, p) = nCx p^x (1-p)^ n-x
n=3; m=5; p=1/2; 1-p=1/2
b(3; 5, 1/2) = 5C3 * 1/2^2 * 1/2^5
= 10 * 1/32
= 10/32
= 5/16
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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select three rainy days out of 5 = 5C3 = 10
probability of 3 rainy and 2 non rainy days = (1/2)^5

Answer = 10 * 1/32 = 5/16
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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probabiliy of raining on 3 days (1/2)^3 x (1/2)^2 = 1/32

the total number of arragements of rain on 3 days and no rain on 2 days is 5!/2!3! = 10

therefore, probability of raining on exactly 3 days during a 5 day week is 10/32 = 5/16
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Joined: 14 Jan 2018
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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how can we solve this kind of problem with different percentages of probability?

So instead of 50 percent lets say 30 percent, I know how to solve this in the "Binomial formula" way, but what about the "n/N" way (N=total possibilities n=rain combinations) what are the total possibilities in the case of 30 percent chances of rain?

thank you
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Re: If the probability of rain on any given day in City X is 50 percent  [#permalink]

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JuliaS wrote:
If the probability of rain on any given day in City X is 50 percent, what is the probability that it rains on exactly 3 days in a 5-day period?

(A) 8/125
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4

P(RRRNN) = (1/2)^5 =1/32

Since RRRNN can be arranged in 5!/(3! x 2!) = 5 x 2 = 10 ways, the overall probability is 10/32 = 5/16.

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