3 yellow balls and 5 black balls

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uptowngirl92: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 19, 2009 9:08 pm; Location: Kolkata,India; Thanked: 7 times; GMAT Score:670

3 yellow balls and 5 black balls

by uptowngirl92 » Tue Sep 29, 2009 12:14 am

A box contains 3 yellow balls and 5 black balls. One by one, every ball is selected at random without replacement. What is the probability that the fourth ball selected is black?
1/4
1/2
1/2
5/8
2/3
3/4

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Source: Beat The GMAT — Problem Solving | Tue Sep 29, 2009 12:14 am

rohan_vus: Master | Next Rank: 500 Posts; Posts: 189; Joined: Thu Apr 03, 2008 2:03 pm; Location: USA; Thanked: 21 times

by rohan_vus » Tue Sep 29, 2009 1:52 am

IMO 5/8 is the answer.

All we need to know is to find the probability of seelcting the fourth ball as black.

Now there could be following 4 situations

1) 3C1*5C2( I select 1 ball fm yellow and 2 from black) .. Probability of fourth black now becomes 3/5 ..(As 3 black remains out of remaining 5 total balls)

2) 3C2*5C1( I select 2 balls fm yellow and 1 from black) .. Probability of fourth black now becomes 4/5 ..(As 4 black remains out of remaining 5 total balls)

3) 3C3( I select 3 balls fm yellow ) .. Probability of fourth black now becomes 5/5 ..(As 5 black remains out of remaining 5 total balls)

4) 5C3( I select 3 balls fm black ) .. Probability of fourth black now becomes 2/5 ..(As 2 black remains out of remaining 5 total balls)

Now lets find the probability of each above sitaution to occur.
Situation (1) can occur in 30 ways out of 56 possible ways ( 8C3 = 56) so probability = 30/56

Similary for situation(2) ,its 15/56
For situation (3), its 1/56
For situation (4) , its 10/56

Now lets combine all situations and get total probability of getting fourth ball as balck.

==> (30/56)*(3/5) + (15/56)*(4/5) + (1/56) *(1) + (10/56)*(2/5)

===35/56 => 5/8 ..So 5/8 is answer

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uptowngirl92: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 19, 2009 9:08 pm; Location: Kolkata,India; Thanked: 7 times; GMAT Score:670

by uptowngirl92 » Tue Sep 29, 2009 8:02 pm

OA is [spoiler]5/8[/spoiler]

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uptowngirl92: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 19, 2009 9:08 pm; Location: Kolkata,India; Thanked: 7 times; GMAT Score:670

by uptowngirl92 » Tue Sep 29, 2009 8:07 pm

SHORTCUT:
The initial probability of drawing a certain ball is the same as the probability of drawing that certain ball at any future point.
Therefore ans: 5/8

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Wed Sep 30, 2009 11:58 am

uptowngirl92 wrote:SHORTCUT:
The initial probability of drawing a certain ball is the same as the probability of drawing that certain ball at any future point.
Therefore ans: 5/8

Awesome!

This kind of thinking is really the key to timing happiness on Test Day.

Let's just use some common sense: if the chance of drawing a black ball were less than 5/8, then the chance of drawing a yellow ball would be greater than 3/8; if the chance of drawing a black ball were greater than 5/8, then the chance of drawing a yellow ball would be less than 3/8? Do either of those scenarios make any sense? No!

Common sense/a bit of logic can be your best friends on the GMAT, don't neglect them!