A stoplight has the following cycle: green for 50 seconds, yellow for 4 seconds, and red for 62 seconds. If you arrive at the light 50 times, predict how many times you will have to stop and wait more than 4 seconds. About 27 times About 50 times About 25 times About 2 times

To solve this problem, we need to analyze the stoplight cycle and determine how many times you will have to stop and wait more than 4 seconds during your 50 arrivals. 1. **Identify the total cycle time of the stoplight:** - Green: 50 seconds - Yellow: 4 seconds - Red: 62 seconds The total cycle time \( T \) is: \[ T = 50 + 4 + 62 = 116 \text{ seconds} \] 2. **Determine the conditions for stopping:** - You will stop if you arrive during the yellow or red light. - The yellow light lasts for 4 seconds, and the red light lasts for 62 seconds. - Therefore, the total time when you will stop is: \[ \text{Stop time} = 4 + 62 = 66 \text{ seconds} \] 3. **Calculate the probability of stopping:** - The probability of stopping when you arrive at the light is the ratio of the stop time to the total cycle time: \[ P(\text{stop}) = \frac{66}{116} \] 4. **Calculate the expected number of stops in 50 arrivals:** - The expected number of times you will stop in 50 arrivals is: \[ E(\text{stops}) = 50 \times P(\text{stop}) = 50 \times \frac{66}{116} \] 5. **Calculate the expected number of stops:** - First, simplify \( \frac{66}{116} \): \[ \frac{66}{116} = \frac{33}{58} \approx 0.569 \] - Now calculate \( E(\text{stops}) \): \[ E(\text{stops}) \approx 50 \times 0.569 \approx 28.45 \] 6. **Determine how many of these stops will be more than 4 seconds:** - You will stop for more than 4 seconds if you arrive during the red light (62 seconds) or during the yellow light (4 seconds) but only if you arrive in the last second of yellow. - The yellow light lasts for 4 seconds, so if you arrive during the first 3 seconds of yellow, you will stop for 4 seconds. If you arrive during the red light, you will stop for 62 seconds. The effective stopping time for more than 4 seconds is: - Red light: 62 seconds - Yellow light: 1 second (the last second of yellow) Therefore, the probability of stopping for more than 4 seconds is: \[ P(\text{stop > 4 seconds}) = \frac{62 + 1}{116} = \frac{63}{116} \] 7. **Calculate the expected number of stops for more than 4 seconds:** \[ E(\text{stops > 4 seconds}) = 50 \times \frac{63}{116} \approx 27.24 \] Based on this calculation, the expected number of times you will have to stop and wait more than 4 seconds is approximately 27 times. Thus, the answer is: **About 27 times**.