Justin is waiting at a bus stop with no knowledge of bus arriving schedule..
But he knows that every 30 minutes, one bus arrives. Find the following probabilities:
(1)He will wait at most 5 minutes =
Incorrect: Your answer is incorrect.

(2))He will wait at least 20 minutes =
Incorrect: Your answer is incorrect.

(3) He will wait at lease 5 minutes but not more than 20 minutes =   '

Given the scenario where a bus arrives every 30 minutes and Justin arrives at a random time, we can model the waiting time as a uniform distribution between 0 and 30 minutes. Let's denote the waiting time as \( W \).

For a uniform distribution \( W \) over the interval \([0, 30]\) minutes, the probability density function \( f_W(w) \) is:
\[ f_W(w) = \frac{1}{30} \quad \text{for} \quad 0 \leq w \leq 30 \]

Now, we can calculate the probabilities for the given scenarios:

### 1. Probability that he will wait at most 5 minutes
This is the probability that \( W \leq 5 \):
\[ P(W \leq 5) = \int_{0}^{5} \frac{1}{30} \, dw = \frac{5}{30} = \frac{1}{6} \approx 0.1667 \]

### 2. Probability that he will wait at least 20 minutes
This is the probability that \( W \geq 20 \):
\[ P(W \geq 20) = \int_{20}^{30} \frac{1}{30} \, dw = \frac{10}{30} = \frac{1}{3} \approx 0.3333 \]

### 3. Probability that he will wait at least 5 minutes but not more than 20 minutes
This is the probability that \( 5 \leq W \leq 20 \):
\[ P(5 \leq W \leq 20) = \int_{5}^{20} \frac{1}{30} \, dw = \frac{15}{30} = \frac{1}{2} = 0.5 \]

Thus, the probabilities are:
1. \( P(W \leq 5) = \frac{1}{6} \approx 0.1667 \)
2. \( P(W \geq 20) = \frac{1}{3} \approx 0.3333 \)
3. \( P(5 \leq W \leq 20) = \frac{1}{2} = 0.5 \)

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the mean waiting time for Justin?
2. How would the probabilities change if the bus arrived every 20 minutes instead of 30?
3. What is the variance of the waiting time distribution?
4. How can we model the waiting time if we know the exact arrival times of buses?
5. How does the probability change if Justin arrives at a specific time instead of randomly?
6. What is the cumulative distribution function (CDF) for Justin's waiting time?
7. How can we calculate the median waiting time for Justin?
8. What if the buses arrived every 45 minutes? How would that affect the waiting probabilities?

**Tip:** In a uniform distribution, the probability of waiting a specific amount of time is directly proportional to the length of the interval considered.

Explore the probabilities of waiting for buses when they arrive every 30 minutes. Understand concepts of probability theory and uniform distribution through practical scenarios. Calculations include waiting times of at most 5 minutes, at least 20 minutes, and between 5 to 20 minutes.

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