A manager at Bird & Brain Investments, I can go home each day only after i ensure that each of my four accountants has finished tallying up their accounts for the day. The office closes at 5:00 PM each day and the tallying work begins immediately after. The time taken by each accountant for tallying is exponentially distributed (of the form C*exp(-rt), where C is a constant and is the rate of the distribution) with expectation 10 minutes and they operate independently of one another. If it takes me 5 minutes to reach home from the office, what is the probability, rounded to the nearest 0.01, that I reach home by 5.30 PM today?

To solve this problem, we need to determine the probability that you finish your work by 5:25 PM, because it takes 5 minutes to get home, and you want to reach home by 5:30 PM. We know that the tallying work of each accountant is exponentially distributed, and the expectation (mean) of the tallying time for each accountant is 10 minutes.

### Step 1: Define the problem setup
Each accountant's tallying time \( T_i \) (for \( i = 1, 2, 3, 4 \)) follows an exponential distribution with an expected value of 10 minutes. The expected value of an exponential distribution is \( \frac{1}{\lambda} \), where \( \lambda \) is the rate parameter of the distribution. From the information given, we know:

\[
\mathbb{E}[T_i] = 10 \text{ minutes} \quad \Rightarrow \quad \lambda = \frac{1}{10} \text{ per minute}.
\]

The total time taken to finish all tallying work is the maximum time out of the four accountants, denoted by \( T_{\text{max}} = \max(T_1, T_2, T_3, T_4) \). We need the probability that this maximum tallying time is less than or equal to 25 minutes, since you have 25 minutes from 5:00 PM to finish tallying and leave the office.

### Step 2: Distribution of the maximum of exponential random variables
For independent exponential random variables, the cumulative distribution function (CDF) of the maximum of \( n \) independent exponential variables is given by the product of the individual CDFs. Specifically, the CDF of the maximum \( T_{\text{max}} \) is:

\[
P(T_{\text{max}} \leq t) = \left( 1 - e^{-\lambda t} \right)^n,
\]
where \( n = 4 \) (since there are four accountants), \( \lambda = \frac{1}{10} \), and \( t = 25 \) minutes.

### Step 3: Calculate the probability
Substitute the values \( \lambda = \frac{1}{10} \), \( n = 4 \), and \( t = 25 \) into the equation:

\[
P(T_{\text{max}} \leq 25) = \left( 1 - e^{-\frac{1}{10} \cdot 25} \right)^4.
\]

First, calculate \( e^{-\frac{25}{10}} = e^{-2.5} \). Let's compute the exact probability.

I'll calculate this now.The probability that you will finish the tallying by 5:25 PM, and thus reach home by 5:30 PM, is approximately **0.71** (rounded to the nearest 0.01).

Do you have any questions or would you like further details?

Here are 5 related questions you might find interesting:
1. How would the probability change if the expected tallying time were 15 minutes instead of 10?
2. What is the expected maximum time for the accountants to finish their work?
3. How does the number of accountants affect the probability of finishing on time?
4. Can you model the probability if the tallying times followed a different distribution, like a normal distribution?
5. How would the probability change if each accountant had a different mean tallying time?

**Tip:** Exponential distributions are often used to model the time between independent events, such as the time taken for individual tasks when there's no memory of previous events.

A manager at Bird & Brain Investments, I can go home each day only after I ensure that each of my four accountants has finished tallying up their accounts for the day. The office closes at 5:00 PM each day and the tallying work begins immediately after. The time taken by each accountant for tallying is exponentially distributed (of the form C*exp(-rt), where C is a constant and r is the rate of the distribution) with expectation 10 minutes and they operate independently of one another. If it takes me 5 minutes to reach home from the office, what is the probability, rounded to the nearest 0.01, that I reach home by 5.30 PM today?

Calculate the probability of finishing tallying work by 5:25 PM with four accountants, each taking an exponentially distributed time to finish with an expectation of 10 minutes. Explore how the distribution of the maximum time impacts the overall probability, using exponential CDFs and related concepts.

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