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the Zebra Crossing is signaled green

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the Zebra Crossing is signaled green

by sanju09 » Tue Dec 03, 2013 2:36 am
Crossing X is operational 24 hours. At 6:00 a.m. on a given day, the Zebra Crossing at the Crossing X is signaled green for 3 minutes. Later on, at each subsequent O'clock till 9:00 p.m. the same day, the Zebra Crossing is signaled green for 1 minute more than the previous permitted number of minutes. After 9:00 p.m. the same day, at each subsequent O'clock till 5:00 a.m. the following day, the Zebra Crossing is signaled green for 2 minute less than the previous permitted number of minutes. A moment is randomly selected from the given period of time. What is the probability that the Zebra Crossing at the Crossing X is signaled green at the selected moment?
A. 1/12
B. 1/9
C. 1/6
D. 1/3
E. 4/11

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by Mathsbuddy » Tue Dec 03, 2013 8:56 am
KEY:
Time -> "Green minutes"

Phase A:
6:00 -> 3
7:00 -> 4
8:00 -> 5
...
N:00 -> N-3
...
20:00 -> 17
21:00 -> 18

Then phase B:
22:00 -> 16
23:00 -> 14
00:00 -> 12
01:00 -> 10
02:00 -> 8
03:00 -> 6
04:00 -> 4
05:00 -> END OF GIVEN TIME PERIOD (not 24 hours!)

Total for Phase A = Sum(1 to 18) - Sum(1 to 2) = 9*19 - 3 = 168
Total for Phase B = 2 * Sum(1 to 8) - 2 = 2 * 4 * 9 - 2 = 70

Total "green minutes" = 168 + 70 = 238

23 hours = 23 * 60 minutes = 1380

Probability, P(green/23h) = 238/1380 = 119/690 which is not on the list.

Therefore, I presume the given time is actually the 24 hour period, so:

P(green/24h) = 240/(24 * 60) = 1/6

Hence ANSWER = (C) 1/6
(and it was not the cunning 'trick' question I thought it to be, relating to the definition of "given hours"!)
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by sanju09 » Wed Dec 04, 2013 12:11 am
Mathsbuddy wrote:KEY:
Time -> "Green minutes"

Phase A:
6:00 -> 3
7:00 -> 4
8:00 -> 5
...
N:00 -> N-3
...
20:00 -> 17
21:00 -> 18

Then phase B:
22:00 -> 16
23:00 -> 14
00:00 -> 12
01:00 -> 10
02:00 -> 8
03:00 -> 6
04:00 -> 4
05:00 -> END OF GIVEN TIME PERIOD (not 24 hours!)

Total for Phase A = Sum(1 to 18) - Sum(1 to 2) = 9*19 - 3 = 168
Total for Phase B = 2 * Sum(1 to 8) - 2 = 2 * 4 * 9 - 2 = 70

Total "green minutes" = 168 + 70 = 238

23 hours = 23 * 60 minutes = 1380

Probability, P(green/23h) = 238/1380 = 119/690 which is not on the list.

Therefore, I presume the given time is actually the 24 hour period, so:

P(green/24h) = 240/(24 * 60) = 1/6

Hence ANSWER = (C) 1/6
(and it was not the cunning 'trick' question I thought it to be, relating to the definition of "given hours"!)
Is there any verbal confusion in the phrase "at each subsequent O'clock till 5:00 a.m", that stops us include 5 O'clock to get a green signal of 2 minutes? May be this way the practice goes forever at the crossing X. If it's really a confusion then why did you include 9:00 p.m. in your first phase work following the same wordings?

I wonder how did I post it in the DS sub=forum, this is the first time it so happened. Dear Moderators please move it to PS sub-forum.

Thanks
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by Mathsbuddy » Thu Dec 05, 2013 4:18 am
sanju09 wrote:
Mathsbuddy wrote:KEY:
Time -> "Green minutes"

Phase A:
6:00 -> 3
7:00 -> 4
8:00 -> 5
...
N:00 -> N-3
...
20:00 -> 17
21:00 -> 18

Then phase B:
22:00 -> 16
23:00 -> 14
00:00 -> 12
01:00 -> 10
02:00 -> 8
03:00 -> 6
04:00 -> 4
05:00 -> END OF GIVEN TIME PERIOD (not 24 hours!)

Total for Phase A = Sum(1 to 18) - Sum(1 to 2) = 9*19 - 3 = 168
Total for Phase B = 2 * Sum(1 to 8) - 2 = 2 * 4 * 9 - 2 = 70

Total "green minutes" = 168 + 70 = 238

23 hours = 23 * 60 minutes = 1380

Probability, P(green/23h) = 238/1380 = 119/690 which is not on the list.

Therefore, I presume the given time is actually the 24 hour period, so:

P(green/24h) = 240/(24 * 60) = 1/6

Hence ANSWER = (C) 1/6
(and it was not the cunning 'trick' question I thought it to be, relating to the definition of "given hours"!)
Is there any verbal confusion in the phrase "at each subsequent O'clock till 5:00 a.m", that stops us include 5 O'clock to get a green signal of 2 minutes? May be this way the practice goes forever at the crossing X. If it's really a confusion then why did you include 9:00 p.m. in your first phase work following the same wordings?

I wonder how did I post it in the DS sub=forum, this is the first time it so happened. Dear Moderators please move it to PS sub-forum.

Thanks
A valid point you make, in that I was inconsistent including and not including the 9pm and 5am respectively. Thank you for pointing that out. The confusion lies in the fact that I don't know if "till N-o'clock" includes the signal that STARTS AT N'o'clock, or if "N-o'clock is the cut-off time, thereby excluding the signal at N-o'clock. The second confusion is that there are 2 given time periods in the question: "Crossing X is operational 24 hours" and "At 6:00 a.m. ... till 5:00 a.m". So this leaves several variations in the potential answer.
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