If you select two cards from a pile of cards numbered 1 to

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RBBmba@2014: Legendary Member; Posts: 944; Joined: Wed May 30, 2012 8:21 am; Thanked: 8 times; Followed by:5 members

If you select two cards from a pile of cards numbered 1 to

by RBBmba@2014 » Thu May 07, 2015 1:57 am

If you select two cards from a pile of cards numbered 1 to 10, what is the probability that the sum of the numbers is less than the average of the pile?

(A) 1/100

(B) 2/45

(C) 2/25

(D) 4/45

(E) 1/10

OA: D

P.S: I did it by considering total outcomes and actual outcomes. Would like to know how it can be solved in other way - I mean, taking 1st card(and counting probabilities) and 2nd card(and counting probabilities) while considering the constraint that sum must be less than average.

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Source: Beat The GMAT — Problem Solving | Thu May 07, 2015 1:57 am

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

by GMATGuruNY » Thu May 07, 2015 2:56 am

RBBmba@2014 wrote:If you select two cards from a pile of cards numbered 1 to 10, what is the probability that the sum of the numbers is less than the average of the pile?

(A) 1/100

(B) 2/45

(C) 2/25

(D) 4/45

(E) 1/10

The 10 cards -- which are composed of the consecutive integers 1 through 10, inclusive -- constitute an EVENLY SPACED SET.
For any evenly spaced set, average = (biggest + smallest)/2.
Thus, the average value of the 10 cards = (10+1)/2 = 5.5.
Implication:
A good outcome occurs when the sum of the 2 cards is 5 or less.

Approach 1:
P = good/all.

All possible outcomes:
Number of options for the 1st card = 10.
Number of options for the 2nd card = 9.
To combine these options, we multiply:
10*9 = 90.

Good outcomes:
1, 2
1, 3
1, 4
2, 1
2, 3
3, 1
3, 2
4, 1.
Total good outcomes = 8.

Thus:
P = 8/90 = 4/45.

The correct answer is D.

Approach 2:
P(1st card is 1 and 2nd card is 2, 3, or 4) = 1/10 * 3/9 = 3/90.
P(1st card is 2 and 2nd card is 1 or 3) = 1/10 * 2/9 = 2/90.
P(1st card is 3 and 2nd card is 1 or 2) = 1/10 * 2/9 = 2/90.
P(1st card is 4 and 2nd card is 1) = 1/10 * 1/9 = 1/90.
Since any of the options above would constitute a good outcome, we ADD the fractions:
3/90 + 2/90 + 2/90 + 1/90 = 8/90 = 4/45.

The correct answer is D.

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talaangoshtari: Master | Next Rank: 500 Posts; Posts: 154; Joined: Wed May 21, 2014 4:29 am; Thanked: 8 times; Followed by:1 members

by talaangoshtari » Fri Jun 05, 2015 10:49 pm

Why does the order matter?

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sanchay: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Wed Jun 26, 2013 11:14 am

by sanchay » Fri Jun 05, 2015 11:47 pm

It doesn't make sense to answer after the OA is given. But still, my guess is D.

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sanchay: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Wed Jun 26, 2013 11:14 am

by sanchay » Fri Jun 05, 2015 11:49 pm

RBBmba@2014 wrote: All possible outcomes:
Number of options for the 1st card = 10.
Number of options for the 2nd card = 9.
To combine these options, we multiply:
10*9 = 90.

Can you please explain this part ?

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

by GMATGuruNY » Sat Jun 06, 2015 2:17 am

talaangoshtari wrote:Why does the order matter?

In my first solution, I calculated the following:
(good orderings)/(all possible orderings).

It is also possible to disregard order and count the following:
(good combinations)/(all possible combinations).

Good combinations:
1, 2
1, 3
1, 4
2, 3.
Total options = 4.

All possible combinations:
From 10 cards, the number of ways to choose a combination of 2 = 10C2 = (10*9)/(2*1) = 45.

Thus:
(good combinations)/(all possible combinations) = 4/45.

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

by GMATGuruNY » Sat Jun 06, 2015 2:26 am

sanchay wrote:
All possible outcomes:
Number of options for the 1st card = 10.
Number of options for the 2nd card = 9.
To combine these options, we multiply:
10*9 = 90.

Can you please explain this part ?

FUNDAMENTAL COUNTING PRINCIPLE:
If one event has m possible outcomes, and a second, independent event has n possible outcomes, then there are m*n total possible ways for the events to happen together.

Problem above:
Since there are 10 possible outcomes for the first card, and 9 possible outcomes for the second card, there are 10*9 = 90 ways that the first two cards can be chosen.

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