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A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and

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KarunMendiratta
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A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   22 Aug 2021, 22:56
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Question Stats:

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A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and 2 torches are defective.
A person takes out two items from the bag at random.

Quantity A
Quantity B
Probability that both are bulbs or both are in working condition
\(\frac{6}{7}\)


A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
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B
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GreNow
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   23 Aug 2021, 13:12
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The probability of selecting 2 bulbs in a row is 2/3 * 9/14 = 3/7 added to the probability of selecting 2 items in working condition which happens to be the same 10/15 * 9/14 = 3/7 and together that's 6/7.
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A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   03 Sep 2021, 20:18
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Explanation:

Defective bulbs = 3
Working bulbs = 7
Defective torches = 2
Working torches = 3

Picking 2 bulbs out of 10 = \(^{10}C_2 = 45\)
Picking 2 torches in working condition out of 3 = \(^3C_2 = 3\)
Picking 1 bulb and 1 torch in working condition = \((^7C_1)(^3C_1) = (7)(3) = 21\)

NOTE: Picking 2 bulbs in working condition have already been included in case 1

Total cases = \(45 + 3 + 21 = 69\)

Total number of ways of picking 2 objects out of 15 = \(^{15}C_2 = 105\)

Col. A: \(\frac{69}{105}\)
Col. B: \(\frac{6}{7}\)

Cross-Multiply the denominators;

Col. A: \(483\)
Col. B: \(630\)

Hence, option B
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   18 Oct 2023, 10:02
I used a shorter method and I am unsure if this works or not. KarunMendiratta and Carcass, please check:

# of working bulbs = 7
# of items to pick from = 15

CHOICE A = P( 2 not defective bulbs) = (7/15) * (6*14) = 1/5 = 20%
CHOICE B = 6/7

B>A
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   18 Oct 2023, 12:46
Expert Reply
Not sure

QA tells us

The probability that both are bulbs or both are in working condition

Both are bulbs OR working conditions.

So you have to include

bulbs
defective bulbs
torches
defective torches
1 bulb and 1 defective torche
1 torche and 1 defective bulb
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   21 Oct 2023, 22:51
GreNow wrote:
c

The probability of selecting 2 bulbs in a row is 2/3 * 9/14 = 3/7 added to the probability of selecting 2 items in working condition which happens to be the same 10/15 * 9/14 = 3/7 and together that's 6/7.


What's wrong with this solution?
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   26 Nov 2023, 12:27
Defective bulbs = 3
Working bulbs = 7
Defective torches = 2
Working torches = 3

both working bulbs or Torches = 7/15x 6/14+ 3/15+2/14= 48/210= 24/105< 6/7. Ans:B
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   20 Oct 2024, 00:32
GreNow wrote:
c

The probability of selecting 2 bulbs in a row is 2/3 * 9/14 = 3/7 added to the probability of selecting 2 items in working condition which happens to be the same 10/15 * 9/14 = 3/7 and together that's 6/7.


why is this incorrect?
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink] New post   20 Oct 2024, 08:47
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Expert Reply
----------Working------Defective-----Total
Bulb--------7---------------3-----------10
Torch-------3---------------2------------5
Total-------10--------------5-----------15

As you can see, there is an overlap between bulbs and working, which should be excluded.
Thus, P = P (both are bulbs) + P (both working) - P (both bulbs AND working).

1. P (both are bulbs):
10/15*9/14=3/7

2. P (both working):
10/15*9/14=3/7

3. P (both bulbs AND working):
7/15*6/14=1/5

P = 3/7+3/7-1/5=23/35 or 0.65

B is 0.85

B is the correct answer.

PS: great question but too tough for the GRE
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Re: A bag contains 10 bulbs and 5 torches, out of which 3 bulbs and [#permalink]
20 Oct 2024, 08:47
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