Hi Everyone,
I encountered a probability question from BTG practice questions. I need your advise in solving this question as i am not able to understand the explanation.
A drawer contains 8 socks, and 2 socks are selected at random without replacement. What is the probability that both socks are black?
(1) The probability is less than 0.2 that the first sock is black.
(2) The probability is more than 0.8 that the first sock is white.
Answer is D
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saurabhkamal1981
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Rahul@gurome
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Given: A drawer contains 8 socks. Say the number of black socks = Bsaurabhkamal1981 wrote:A drawer contains 8 socks, and 2 socks are selected at random without replacement. What is the probability that both socks are black?
(1) The probability is less than 0.2 that the first sock is black.
(2) The probability is more than 0.8 that the first sock is white.
Thus the probability that both socks are black = (The probability that the first sock is black)*(The probability that the second sock is black) = (B/8)*((B - 1)/7)
Therefore if we know B, we can determine the probability.
Statement 1: The probability is less than 0.2 that the first sock is black.
Thus, (B/8) < 0.2 => B < 1.6
As B must be a non-negative integer, possible values of B are 0 and 1.
- 1. If B = 0: Required probability = 0
1. If B = 1: Required probability = 0
Statement 2: The probability is more than 0.8 that the first sock is white.
Say number of white socks = W, then (W/8) > 0.6 => W > 6.4
As W must be a non-negative integer, possible values of W are 7 and 8. Thus the number of black socks is either 0 or 1. The rest of the analysis is same as statement 1.
Sufficient.
The correct answer is D.
Rahul Lakhani
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Gurome, Inc.
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On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
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Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
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saurabhkamal1981
- Senior | Next Rank: 100 Posts
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- Joined: Sun Nov 08, 2009 5:03 am
Statement 1: The probability is less than 0.2 that the first sock is black.
Thus, (B/8) < 0.2 => B < 1.6
As B must be a non-negative integer, possible values of B are 0 and 1.
1. If B = 0: Required probability = 0
1. If B = 1: Required probability = 0
Sufficient.
Statement 2: The probability is more than 0.8 that the first sock is white.
Say number of white socks = W, then (W/8) > 0.6 => W > 6.4
As W must be a non-negative integer, possible values of W are 7 and 8. Thus the number of black socks is either 0 or 1. The rest of the analysis is same as statement 1.
Rahul thank you very much for your help and really appreciate for your explanation. I have understood the explanation but confused with two things.
First, as you have mentioned in statement 1 that possible values of B are 0 and 1, this is fine, i am able to understand this part, but what about the following:-
1. If B = 0: Required probability = 0
1. If B = 1: Required probability = 0
Can you please elaborate the above bold part.
Second, in statement 2 possible values of W are 7 and 8, this is okay, i am able to understand this part also, but what about this one:-
number of black socks is either 0 or 1. --> how we can conclude that number of black socks is either 0 or 1
Can you please explain these two things.
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Rahul@gurome
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As I have mentioned in the "Given" analysis, the probability that both socks are black = (The probability that the first sock is black)*(The probability that the second sock is black) = (B/8)*((B - 1)/7)saurabhkamal1981 wrote:First, as you have mentioned in statement 1 that possible values of B are 0 and 1, this is fine, i am able to understand this part, but what about the following:-
1. If B = 0: Required probability = 0
1. If B = 1: Required probability = 0
Can you please elaborate the above bold part.
This is because...
- Probability that the first sock is black = (B/8)
Probability that the second sock is black = ((B - 1)/7)
Now put the possible values of B in the expression. For B = 0 and B = 1, the probability is zero!
(This can be shown without introducing any formula. If the number of black socks is zero then the probability that both socks are black is obviously zero as there is no black socks at all! If the number of black socks is one then also the probability that both socks are black is obviously zero as after the first pick there is no black socks!)
Remember that total number of socks is 8. Now there are either 7 or 8 white socks.saurabhkamal1981 wrote:Second, in statement 2 possible values of W are 7 and 8, this is okay, i am able to understand this part also, but what about this one:-
number of black socks is either 0 or 1. --> how we can conclude that number of black socks is either 0 or 1
If there are 8 white socks then all the socks are white => No black socks.
If there are 7 white socks then only one sock is non-white => Number of black socks = 1 (If the remaining sock is black) or Number of black socks = 0 (If the remaining sock is of some other color)
Thus number of black sock is either 0 or 1.
Rahul Lakhani
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
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saurabhkamal1981
- Senior | Next Rank: 100 Posts
- Posts: 31
- Joined: Sun Nov 08, 2009 5:03 am
Rahul@gurome wrote:
As I have mentioned in the "Given" analysis, the probability that both socks are black = (The probability that the first sock is black)*(The probability that the second sock is black) = (B/8)*((B - 1)/7)
This is because...Hope this is okay with you.
- Probability that the first sock is black = (B/8)
Probability that the second sock is black = ((B - 1)/7)
Now put the possible values of B in the expression. For B = 0 and B = 1, the probability is zero!
(This can be shown without introducing any formula. If the number of black socks is zero then the probability that both socks are black is obviously zero as there is no black socks at all! If the number of black socks is one then also the probability that both socks are black is obviously zero as after the first pick there is no black socks!)
Okay got it. Thank you very much Rahul and really appreciate for your help.Rahul@gurome wrote:Remember that total number of socks is 8. Now there are either 7 or 8 white socks.
If there are 8 white socks then all the socks are white => No black socks.
If there are 7 white socks then only one sock is non-white => Number of black socks = 1 (If the remaining sock is black) or Number of black socks = 0 (If the remaining sock is of some other color)
Thus number of black sock is either 0 or 1.
