A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take 0.8,1.6 ,2.4 , or  3.2 hours. The different types of malfunctions occur at the same frequency.

If required, round your answers to two decimal places.                                                                                            A service call has just come in, but the type of malfunction is unknown. It is 3 P.M. and service technicians usually get off at  5 P.M. What is the probability the service technician will have to work overtime to fix the machine today?

To solve this problem, let's break it down step by step:

### Step 1: Determine what overtime means in terms of hours worked.
The technician gets off work at 5 PM, and the current time is 3 PM. This means the technician has 2 hours (from 3 PM to 5 PM) to finish the service call. If the service call takes more than 2 hours, the technician will have to work overtime.

### Step 2: Identify the possible service times.
The possible durations for the service call are: 
- 0.8 hours
- 1.6 hours
- 2.4 hours
- 3.2 hours

### Step 3: Determine which times lead to overtime.
The technician will work overtime if the service call duration exceeds 2 hours. The possible durations that cause overtime are:
- 2.4 hours
- 3.2 hours

### Step 4: Calculate the probability of overtime.
The problem states that all malfunctions occur with the same frequency, so each duration is equally likely. Since there are 4 possible durations, the probability for each one is:

\[
P(\text{each duration}) = \frac{1}{4}
\]

Now, the durations leading to overtime are 2.4 hours and 3.2 hours. So the probability of working overtime is the sum of the probabilities for these two durations:

\[
P(\text{overtime}) = P(2.4 \text{ hours}) + P(3.2 \text{ hours}) = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = 0.5
\]

### Final Answer:
The probability that the technician will have to work overtime is **0.5** or **50%**.

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Would you like further details or clarifications on any part of this solution?

Here are 5 related questions:
1. How would the result change if the technician got off at 6 PM instead of 5 PM?
2. What if one type of malfunction was more frequent than the others?
3. How can we compute the expected value of service time?
4. How would the probability change if the time available was reduced to 1.5 hours?
5. Can the probability be different if we had more possible malfunction durations?

**Tip:** When dealing with equally likely outcomes, the probability of an event is simply the ratio of favorable outcomes to the total number of possible outcomes.

A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take 0.8, 1.6, 2.4, or 3.2 hours. The different types of malfunctions occur at the same frequency. If required, round your answers to two decimal places. A service call has just come in, but the type of malfunction is unknown. It is 3 P.M. and service technicians usually get off at 5 P.M. What is the probability the service technician will have to work overtime to fix the machine today?

Learn how to calculate the probability that a service technician will have to work overtime when servicing machines. This problem explores uniform distribution and basic probability concepts, determining the likelihood of a technician exceeding available work hours. Suitable for students in Grades 9-12.

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