Help with this problem

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knight247: Legendary Member; Posts: 504; Joined: Tue Apr 19, 2011 1:40 pmFollowed by: 11 members; Thanked: 114 times

by knight247 » Mon Jun 20, 2011 1:22 am

In Rwanda, the chance for rain on any given day is 50%. What is the probability that it rains on 4 out of 7 consecutive days in Rwanda?

a) 4/7
b) 3/7
c) 35/128
d) 4/28
e) 28/135

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Source: Beat The GMAT — Problem Solving | Mon Jun 20, 2011 1:22 am

winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Mon Jun 20, 2011 1:47 am

Chances of raining = 50% therefore there are two options: either it rains or it does not rain.
Therefore the total options are 2^7 = 128 ways.

Now, say it rains on 4 days, so the number of ways it can rain on 4 given days = 7C4 = 7!/4! x 3! =35 ways.

Therefore probability is 35/128.

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knight247: Legendary Member; Posts: 504; Joined: Tue Apr 19, 2011 1:40 pmFollowed by: 11 members; Thanked: 114 times

by knight247 » Mon Jun 20, 2011 2:23 am

35/128 is correct. i have an issue though with 'the number of ways it can rain on 4 given days'
We have 4 rain days and 3 non rain days. In figures it would look as follows:

R R R R N N N Where 'R' stands for rain day and 'N' for non rain day

which can be arranged as

N N N R R R R
R R N N R R N
R N R N R N R
.
.
.
ETC

Shouldnt 'the number of ways it can rain on 4 given days' be 7P7 = 7!/(4!*3!) ???? Which btw is also equal to 35. Hope U can clarify this!!

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Mon Jun 20, 2011 2:30 am

Firstly, 7P7 is not equal to 35.
7P7 = 7!/(7-7)! = 7!/0! = 7!/1 = 7! ways.

Secondly, you should use permutation formula only when the order of the occurrence of an event is important.In this case however it does not matter when it rains, it can rain on any 4 days.

However, if the problem specified that it should rain on every alternate day then that would be a different scenario!

Last edited by winniethepooh on Mon Jun 20, 2011 3:01 am, edited 1 time in total.

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knight247: Legendary Member; Posts: 504; Joined: Tue Apr 19, 2011 1:40 pmFollowed by: 11 members; Thanked: 114 times

by knight247 » Mon Jun 20, 2011 2:40 am

7! is definitely not equal to 35. In case of R R R R N N N ....7P7 would be 7!/(4!*3!) because the Rs are repeated 4 times and the Ns are repeated 3 times. Which is equal to 35. Hope i clarified the point i'm trying to make.

And in this case isn't order important? R R R R N N N isn't the same as N N N R R R R

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Mon Jun 20, 2011 2:59 am

You're going on the wrong track buddy. Remember, always the formula for Permutations is: nPr = n!/r!x(n-r)!
For Combinations : nCr = n!/(n-r)!
Another point to remember is that a permutation of a certain number will always be higher than the combination of the same number. Hence, never can 7C7 be equal to 7P7!

Also,in this problem it doesn't matter if it rains in this pattern NNNNRRR versus in this pattern RRRRNNN.

Last edited by winniethepooh on Mon Jun 20, 2011 6:22 am, edited 1 time in total.

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Mon Jun 20, 2011 3:24 am

Sorry, last pattern of RNRNRNR wont work as the problem says 4 consecutive days.

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Surev: Senior | Next Rank: 100 Posts; Posts: 36; Joined: Thu May 26, 2011 7:19 am; Thanked: 3 times

by Surev » Mon Jun 20, 2011 6:50 am

Winnie - you have the right solution but the question does not say 4 consecutive days. It says any 4 days out of 7 consecutive days. That is the reason we take 7C4. If it had been 4 consecutive days, the approach would be different.

Another way of looking at it would be as that of rearranging the letters in a word to form a new word. Suppose the word is RRRRNNN. There are a total of 7 letters of which 4 are identical and 3 are identical. Therefore the number of ways to rearrange will be 7!/(4!*3!) which is same as 7C4.

Hope this helps.

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Mon Jun 20, 2011 3:39 pm

Mt bad, read it wrong!

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