Final answer:
The probabilities were calculated using the length of the time intervals over the total possible duration for a vaping cartridge which lasts uniformly between 5 and 8 hours. The expected duration is the midpoint of the range.
Step-by-step explanation:
The amount of time a vaping cartridge lasts is uniformly distributed between 5 hours and 8 hours. To find the probabilities for the various intervals, we will follow these steps:
- First, we understand that a uniform distribution means that every outcome in the range has an equal chance of occurring, so the probability is the length of the time interval divided by the total length of time.
- The total duration in which the cartridge can last is 3 hours (from 5 to 8 hours), which is 180 minutes.
Answers to the questions:
- Probability it will last less than 7 hours and 15 minutes: 7 hours and 15 minutes is equivalent to 435 minutes. So, the probability = (435 - 300) / 180 = 0.75 or 75%.
- Probability it will last more than 6 hours and 45 minutes: 6 hours and 45 minutes is 405 minutes. So, the probability = (480 - 405) / 180 = 0.4167 or 41.67%.
- Probability it will last between 5 hours and 30 minutes, and 6 hours and 50 minutes: 5 hours and 30 minutes is 330 minutes, and 6 hours and 50 minutes is 410 minutes. So, the probability = (410 - 330) / 180 = 0.4444 or 44.44%.
- The expected time a vaping cartridge will last is the midpoint of the uniform distribution range, which is (5 hours + 8 hours) / 2 = 6.5 hours or 6 hours and 30 minutes.