Final answer:
The student's questions pertain to computing probabilities for waiting times and intervals between events using the uniform and exponential probability distributions, as well as the Poisson distribution for discrete events.
Step-by-step explanation:
The questions provided predominantly revolve around calculating probabilities for different scenarios using continuous probability distributions, specifically the uniform and exponential distributions. As these questions are mathematical in nature, they require understanding of the underlying principles of probability theory and distribution functions.
Probability of Waiting Time
For instance, the question regarding the probability that a person waits fewer than 12.5 minutes for a bus uniformly distributed between zero and 15 minutes is solved by finding the area under the uniform distribution curve from zero to 12.5 minutes. Similarly, the exponentially distributed times between events like calls, website visits, or patient arrivals can be analyzed using the exponential probability density function to find probabilities and percentiles.
Computing Exponential Probabilities
Calculating the probability that fewer than 20 calls occur within an hour when on average four calls occur per minute requires the use of the Poisson distribution. Additionally, computing the likelihood of the next call occurring within a certain time after a known interval has passed involves the memoryless property of the exponential distribution.
To solve these types of probability problems, we would typically identify the relevant probability distribution, use the density function to find probabilities for continuous variables, and use cumulative distribution functions for discrete intervals.