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?

Answer 1

Hint: Here we are provided with three traffic lights changing at certain seconds of time we will find the length of one time cycle. And with the help of the timing we will find the total possibility of color changing, and hence with the help of the following formula we will find the probability required.

Formula used:
\[{\text{Probability of an event happening}} = \dfrac{{{\text{Number of favorite outcomes}}}}{{{\text{Total number of outcomes}}}}\]

Complete step by step solution:
It is given that a traffic light runs repeatedly through the following time cycle: green for \[30\] seconds, then yellow for \[3\] seconds, and then red for \[30\] seconds.
That is the traffic light runs through a \[63\] seconds to complete one time cycle.
Also it is given that Leah picks a random 3 second time interval to watch the light.
We should find the probability that the color changes while Leah is watching.
Green light changes after 30 seconds and after three seconds yellow light changes that is at the thirty third second yellow changes and after another 30 seconds the red light changes.
Now we let the time \[t = 0\] reference the moment it turns green,
The light changes at three different times:\[t = 30\], \[t = 33\], and \[t = 63\]
This means that the light will change if the Leah watches in the intervals\[\left[ {27,30} \right]\], \[\left[ {30,33} \right]\] or \[\left[ {60,63} \right]\]
Since every interval has 3 seconds and there are 3 intervals there will be a chance of 9 seconds for Leah.
This gives a total of \[9\] seconds out of \[63\]
$\Rightarrow$\[{\text{Probability of an Leah watching the color change}} = \dfrac{{{\text{Number of seconds she watches}}}}{{{\text{Total seconds}}}}\]\[ = \dfrac{9}{{63}} = \dfrac{1}{7}\]

$\therefore$ The probability that Leah watching the color change is \[\dfrac{1}{7}\]

Note: Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability. Don’t get confused while counting the number of favourable outcomes, it is just the desired number of outcomes in our experiment.