rain prob

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ruplun: Master | Next Rank: 500 Posts; Posts: 232; Joined: Fri Jun 18, 2010 7:09 amFollowed by: 2 members; Thanked: 1 times

by ruplun » Mon Aug 01, 2011 11:35 am

probability of raining on any day in a week is 50 % .what is the prob that it rains exactly 3 days in a span of 5 days?

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Source: Beat The GMAT — Problem Solving | Mon Aug 01, 2011 11:35 am

edge: Senior | Next Rank: 100 Posts; Posts: 49; Joined: Sun Jun 26, 2011 5:45 pm; Thanked: 2 times

by edge » Mon Aug 01, 2011 11:58 am

There are â�µCâ‚ƒ = 10 different 3-day combinations from a group of 5 days.

Probability of raining first 3 days out of 5 = (0.5)^3 * (1 - 0.5)^2 = 0.03125. We know that there are 10 such outcomes.

Therefore, actual probability = 10 * 0.03125 = 0.3125.

I could be way off the mark on this; can someone confirm? Experts?

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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 Aug 01, 2011 12:37 pm

It can rain in 5C3 ways in 5 days = 10 ways.
the probability that it rains on any three days = 1/2 * 1/2 * 1/2 = 1/8 ways
thw probability that it does not rain on any 2 days = 1/2 * 1/2 = 1/4
Total probability of raining on exactly any three days = 10 * 1/8 * 1/4 = 5 /16 = .3125

@ edge: Please don't convert fractions into decimals. It will make calculations harder for you.

Last edited by winniethepooh on Tue Aug 02, 2011 12:47 am, edited 1 time in total.

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pemdas: Legendary Member; Posts: 1084; Joined: Fri Apr 15, 2011 2:33 pmFollowed by: 21 members; Thanked: 158 times

by pemdas » Mon Aug 01, 2011 1:08 pm

if we calculate the compliment probability, i.e. it doesn't rain exactly two days in a span of five days, then we have 5C2=10 combination sets,
we get P(2 days doesn't rain)=((1/2)^5 )*10=5/16 which means P(3 days rain)

winniethepooh wrote:It can rain in 5C3 ways in 5 days = 10 ways.
the probability that it rains on any three days = 1/2 * 1/2 * 1/2 = 1/8 ways
thw probability that it does not rain on any 2 days = 1/2 * 1/2 = 1/4
Total probability of raining on exactly any three days = 10 * 1/8 * 1/4 = 5 /16 = .3125

@ edge: Please don't convert fractions into decimals. It will make calculations harder for you.
You probably forgot to multiply your result by 10.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Aug 01, 2011 2:12 pm

ruplun wrote:probability of raining on any day in a week is 50 % .what is the prob that it rains exactly 3 days in a span of 5 days?

P(exactly n times) = P(one way) * total possible ways.

Let R = rain and N = no rain.

P(one way):
One way to get exactly 3 days of rain is to have rain on the first 3 days but not on the last 2 days.
P(RRRNN) = 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32.

Total possible ways:
Any arrangement of the letters RRRNN will yield exactly 3 days of rain.
Thus, to account for all the ways to get exactly 3 days of rain, the result above needs to be multiplied by the number of ways to arrange RRRNN.
Number of ways to arrange RRRNN = 5!/(3!2!) = 10.

Multiplying the results above, we get:
P(exactly 3 days of R) = 10 * 1/32 = 10/32 = 5/16.

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edge: Senior | Next Rank: 100 Posts; Posts: 49; Joined: Sun Jun 26, 2011 5:45 pm; Thanked: 2 times

by edge » Mon Aug 01, 2011 2:33 pm

winniethepooh wrote:@ edge: Please don't convert fractions into decimals. It will make calculations harder for you. You probably forgot to multiply your result by 10.

Huh? I multiplied by 10...

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ruplun: Master | Next Rank: 500 Posts; Posts: 232; Joined: Fri Jun 18, 2010 7:09 amFollowed by: 2 members; Thanked: 1 times

by ruplun » Mon Aug 01, 2011 11:40 pm

The explanation and solution provided by GMATGuruNY is very good ... it makes the reasoning and picture far clearer....

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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 » Tue Aug 02, 2011 12:49 am

edge wrote:
winniethepooh wrote:@ edge: Please don't convert fractions into decimals. It will make calculations harder for you. You probably forgot to multiply your result by 10.
Huh? I multiplied by 10...

Oops, my bad!! I missed that!
Edited above.

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