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On your way to work, you have to drive through a busy juction where you may be stopped at traffic lights. The cycle of the traffic light is 2 minutes of green followed by 3 minutes of red. What is the expected delay in the journey, if you arrive at the junction at a random time uniformly distributed over the whole 5 minute cycle is?
- a)9/10 minute
- b)3/5 minute
- c)3/10 minute
- d)3/2 minute
Correct answer is option 'D'. Can you explain this answer?
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On your way to work, you have to drive through a busy juction where yo...
Delay in journey is denoted by x then its probability density function.
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On your way to work, you have to drive through a busy juction where yo...
Understanding the Problem:
We are given that the cycle of the traffic light is 2 minutes of green followed by 3 minutes of red. We need to find the expected delay in the journey if we arrive at the junction at a random time uniformly distributed over the whole 5-minute cycle.
Solution:
To solve this problem, we can consider the possible scenarios for the arrival time at the junction and calculate the expected delay for each scenario.
Scenario 1: Arrive during the green phase
If we arrive during the 2-minute green phase, there will be no delay as we can pass through the junction without stopping.
Scenario 2: Arrive during the red phase
If we arrive during the 3-minute red phase, we will have to wait for the entire red phase before the light turns green. In this case, the delay will be 3 minutes.
Scenario 3: Arrive just before the green phase
If we arrive just before the green phase, the delay will depend on the remaining time in the red phase. Since the arrival time is uniformly distributed over the 5-minute cycle, the probability of arriving just before the green phase is 2/5 (2 minutes before the green phase starts).
In this case, the expected delay can be calculated as the average of the possible delays. The possible delays are 0 minutes if the green phase starts immediately after arrival and 3 minutes if the arrival is at the beginning of the red phase. Therefore, the expected delay for this scenario is (0 + 3)/2 = 3/2 minutes.
Calculating the Expected Delay:
To calculate the overall expected delay, we need to consider the probabilities of each scenario happening.
- The probability of arriving during the green phase is 2/5 (2 minutes out of 5 minutes).
- The probability of arriving during the red phase is 3/5 (3 minutes out of 5 minutes).
Therefore, the expected delay can be calculated as:
(Probability of Scenario 1 * Delay in Scenario 1) + (Probability of Scenario 2 * Delay in Scenario 2) + (Probability of Scenario 3 * Delay in Scenario 3)
= (2/5 * 0) + (3/5 * 3) + (2/5 * 3/2)
= 0 + 9/5 + 3/5
= 12/5
= 2 2/5 minutes
Conclusion:
The expected delay in the journey, if you arrive at the junction at a random time uniformly distributed over the whole 5-minute cycle, is 2 2/5 minutes. Therefore, the correct option is D) 3/2 minutes.
We are given that the cycle of the traffic light is 2 minutes of green followed by 3 minutes of red. We need to find the expected delay in the journey if we arrive at the junction at a random time uniformly distributed over the whole 5-minute cycle.
Solution:
To solve this problem, we can consider the possible scenarios for the arrival time at the junction and calculate the expected delay for each scenario.
Scenario 1: Arrive during the green phase
If we arrive during the 2-minute green phase, there will be no delay as we can pass through the junction without stopping.
Scenario 2: Arrive during the red phase
If we arrive during the 3-minute red phase, we will have to wait for the entire red phase before the light turns green. In this case, the delay will be 3 minutes.
Scenario 3: Arrive just before the green phase
If we arrive just before the green phase, the delay will depend on the remaining time in the red phase. Since the arrival time is uniformly distributed over the 5-minute cycle, the probability of arriving just before the green phase is 2/5 (2 minutes before the green phase starts).
In this case, the expected delay can be calculated as the average of the possible delays. The possible delays are 0 minutes if the green phase starts immediately after arrival and 3 minutes if the arrival is at the beginning of the red phase. Therefore, the expected delay for this scenario is (0 + 3)/2 = 3/2 minutes.
Calculating the Expected Delay:
To calculate the overall expected delay, we need to consider the probabilities of each scenario happening.
- The probability of arriving during the green phase is 2/5 (2 minutes out of 5 minutes).
- The probability of arriving during the red phase is 3/5 (3 minutes out of 5 minutes).
Therefore, the expected delay can be calculated as:
(Probability of Scenario 1 * Delay in Scenario 1) + (Probability of Scenario 2 * Delay in Scenario 2) + (Probability of Scenario 3 * Delay in Scenario 3)
= (2/5 * 0) + (3/5 * 3) + (2/5 * 3/2)
= 0 + 9/5 + 3/5
= 12/5
= 2 2/5 minutes
Conclusion:
The expected delay in the journey, if you arrive at the junction at a random time uniformly distributed over the whole 5-minute cycle, is 2 2/5 minutes. Therefore, the correct option is D) 3/2 minutes.
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On your way to work, you have to drive through a busy juction where yo...
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On your way to work, you have to drive through a busy juction where you may be stopped at traffic lights. The cycle of the traffic light is 2 minutes of green followed by 3 minutes of red. What is the expected delay in the journey, if you arrive at the junction at a random time uniformly distributed over the whole 5 minute cycle is?a)9/10 minuteb)3/5 minutec)3/10 minuted)3/2 minuteCorrect answer is option 'D'. Can you explain this answer?
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