A traffic light runs repeatedly through the following cycle: green for

Question

A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?

(A)  \frac{1}{63}

(B)  \frac{1}{21}

(C)  \frac{1}{10}

(D)  \frac{1}{7}

(E)  \frac{1}{3}

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imeanup · Answer

t = 30 + 3 + 30 = 63

3 second each interval and 3 choices   3 	imes 3 = 9

P = \dfrac{9}{63} = \dfrac{1}{7}

D

A_Nishith · Answer

This cycle repeats indefinitely, and the total duration of one full cycle is:

30 seconds (Green)+3 seconds (Yellow)+30 seconds (Red)=63 seconds

Leah picks a random three-second interval to watch the light. We need to find the probability that the light changes color during her three-second interval.

When does the light change? 
-> From Green to Yellow (at 30 seconds)
-> From Yellow to Red (at 33 seconds)
-> From Red to Green (at 63 seconds, which is also 0 seconds of the next cycle)
 So, the light changes color at the following seconds in the cycle: 30, 33, and 63.

=> For a three-second interval to include a color change:

The interval must start within 3 seconds before the light change. Specifically:
If Leah starts watching at 28 to 30 seconds, she will see the change from Green to Yellow.
If Leah starts watching at 31 to 33 seconds, she will see the change from Yellow to Red.
If Leah starts watching at 61 to 63 seconds, she will see the change from Red to Green.
Each of these intervals lasts for 3 seconds.

Total Favorable Intervals: 
There are 3 intervals during which Leah would see a color change:

28 to 30 seconds (3 seconds)
31 to 33 seconds (3 seconds)
61 to 63 seconds (3 seconds)
So,  there are 9 seconds out of the total 63 seconds during which she would observe a color change.

Probability= {Number of favorable seconds/ Total seconds in the cycle} = 9/63 = 1/7

Answer: D
​

ribbons · Answer

Another approach

There are 10-1-10 intervals of 3seconds each

In which the color changes 3 times, from green to yellow, yellow to red, red to green

So we have 3 intervals of colour changes out of 21 total intervals hence 3/21=1/7

Dhananjayrai · Answer

The question stem asks for the possibilities wherein the light changes during the three - second interval. So shouldn’t it be -> 28 sec - 30 sec(1); 29 sec - 31 sec(2) ; 31 sec - 33 sec(3) ; 32 sec - 34 sec(4) ; 61 sec - 63 sec(5) ; 62 sec - 1 sec(6)( again green). Total 6 such intervals. In all these intervals, there will be a change in light.

Posted from my mobile device

bumpbot · Answer

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A traffic light runs repeatedly through the following cycle: green for

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