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# A bag contains 15 wool scarves, exactly one of which is red and exactl

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Manager
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A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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17 Mar 2015, 10:20
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60% (01:51) correct 40% (01:59) wrong based on 130 sessions

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A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35
b) 1/15
c) 6/35
d) 13/70
e) 1/5
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 43787
A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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17 Mar 2015, 10:29
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ynaikavde wrote:
A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35
b) 1/15
c) 6/35
d) 13/70
e) 1/5

So, we need 1 red out of 1, and 2 (some other color but red and green) out of 13.

$$P = \frac{C^1_1*C^2_{13}}{C^3_{15}}=\frac{78}{455}=\frac{6}{35}$$,

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Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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18 Mar 2015, 22:34
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Hi ynaikavde,

The "math" behind this question can also be handled as a Permutation. Here's how:

We have 15 scarves: 1 red, 1 green and 13 other. We're asked for the probability of grabbing 3 scarves that include the 1 red but NOT the 1 green scarf.

Even though the question states that we are supposed to grab all 3 scarves simultaneously, we can still refer to them as 1st, 2nd and 3rd. There are 3 ways to get what we *want*

R = Red
O = Other

ROO
ORO
OOR

The probability of getting ROO = (1/15)(13/14)(12/13) = 2/35

The probability of getting ORO = (13/15)(1/14)(12/13) = 2/35

The probability of getting OOR = (13/15)(12/14)(1/13) = 2/35

Total ways = 2/35 + 2/35 + 2/35 = 6/35

[Reveal] Spoiler:
C

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Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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19 Mar 2015, 01:29
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KUDOS
Ans- C

Sample space= $$15C3$$
No. of possible outcomes when one red scarf to be selected and green scarf is not to be selected= $$13C2$$

Required probability = $$\frac{13C2}{15C3}$$
= $$\frac{(13*6)}{(5*7*13)}$$
= $$\frac{6}{35}$$
Manager
Joined: 25 Apr 2013
Posts: 66
Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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19 Mar 2015, 05:42
Probability = number of favorable outcomes/total number of possible outcomes

Total number of possible outcomes = C(15,3)

Number of favorable outcomes = Number of ways in which one red scarf and two non-green scarves can be drawn
Number of favorable outcomes = 1*C(13,2)

This is because there is only one way in which one red scarf can be drawn, while there are C(13,2) ways in which two non-green scarves can be drawn, because we have to draw any two out of the 13 remaining (since red one is already drawn and green one we don't have to draw).

So, probability = C(13,2)/C(15,3)=6/35
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Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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29 May 2016, 15:24
Bunuel wrote:
ynaikavde wrote:
A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35
b) 1/15
c) 6/35
d) 13/70
e) 1/5

So, we need 1 red out of 1, and 2 (some other color but red and green) out of 13.

$$P = \frac{C^1_1*C^2_{12}}{C^3_{15}}=\frac{78}{455}=\frac{6}{35}$$,

You mean C, correct?
Senior Manager
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A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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29 May 2016, 22:45
ynaikavde wrote:
A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35
b) 1/15
c) 6/35
d) 13/70
e) 1/5

Theory:

To Find Probability when a red scarf is chosen but a green scarf is not chosen.
First Find probability when Red scarf is necessarily chosen. say this is (1)
Then Find Probability when both of them are chosen together. Say this is (2)
(1) - (2) will give the probability when Red is chosen but Green is not chosen

[ Theory: P(AUB`) = P(A) - P(A intersection B)]

Probability of selecting 3 scarves such that a Red scarf is necessarily selected P(R)
= $$\frac{14C2}{15C3}$$ (We always choose the red scarf, no restrictions on other two)
= ($$\frac{14!}{12!2!}$$) divided by ($$\frac{15!}{12!3!}$$)
= $$\frac{14!12! 3!}{12!2! 15!}$$
= $$\frac{3}{15}$$
= $$\frac{1}{5}$$

Probability of selecting 3 scarves such that One of them is Red and One is a Green scarf
= $$\frac{13C1}{15C3}$$ (We choose the red scarf, We choose the green scarf, then out of remaining 13 we choose remaining 1 scarf)
= ($$\frac{13!}{12!}$$) divided by ($$\frac{15!}{12!3!}$$)
= $$\frac{13!12! 3!}{15!12!}$$
= $$\frac{13!3!}{15!}$$
= $$\frac{6}{15*14}$$
= $$\frac{1}{35}$$

Now we have to find the Probability that a Red Scarf is selected but a Green scarf is not selected.

This will be [ Probability when a Red Scarf is definitely chosen - Probability when a Red Scarf+Green Scarf are chosen together]
= $$\frac{1}{5}$$ - $$\frac{1}{35}$$
= $$\frac{7}{35}$$ - $$\frac{1}{35}$$
= $$\frac{6}{35}$$

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Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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16 Jun 2016, 13:11
1
KUDOS
Bunuel wrote:
ynaikavde wrote:
A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35
b) 1/15
c) 6/35
d) 13/70
e) 1/5

So, we need 1 red out of 1, and 2 (some other color but red and green) out of 13.

$$P = \frac{C^1_1*C^2_{12}}{C^3_{15}}=\frac{78}{455}=\frac{6}{35}$$,

Hi Bunuel,

In the solution, you mentioned 12c2 but it has to be 13c2 instead, only then we can get 78.
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Joined: 02 Sep 2009
Posts: 43787
Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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17 Jun 2016, 00:56
msk0657 wrote:
Bunuel wrote:
ynaikavde wrote:
A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35
b) 1/15
c) 6/35
d) 13/70
e) 1/5

So, we need 1 red out of 1, and 2 (some other color but red and green) out of 13.

$$P = \frac{C^1_1*C^2_{12}}{C^3_{15}}=\frac{78}{455}=\frac{6}{35}$$,

Hi Bunuel,

In the solution, you mentioned 12c2 but it has to be 13c2 instead, only then we can get 78.

Edited the typo. Thank you.
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Re: A bag contains 15 wool scarves, exactly one of which is red and exactl [#permalink]

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25 Sep 2017, 10:19
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Re: A bag contains 15 wool scarves, exactly one of which is red and exactl   [#permalink] 25 Sep 2017, 10:19
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