10 items have failed out of 1010 put on test during a period of 5000 hours, calculate the failure rate reliability questions solve 

To calculate the failure rate and other reliability-related metrics, we need to consider the following:

### 1. **Failure Rate (λ):**
The failure rate is calculated using the formula:

\[
\lambda = \frac{\text{Number of Failures}}{\text{Total Operating Time}}
\]

**Total Operating Time** is the product of the number of items and the total time period:

\[
\text{Total Operating Time} = \text{Number of Items} \times \text{Test Period}
\]

Given:
- Number of items = 1010
- Test period = 5000 hours
- Number of failures = 10

Thus:

\[
\text{Total Operating Time} = 1010 \times 5000 = 5,050,000 \text{ hours}
\]

Now, calculate the failure rate:

\[
\lambda = \frac{10}{5,050,000} \approx 1.98 \times 10^{-6} \text{ failures per hour}
\]

### 2. **Mean Time Between Failures (MTBF):**
The MTBF is the inverse of the failure rate:

\[
\text{MTBF} = \frac{1}{\lambda} \approx \frac{1}{1.98 \times 10^{-6}} \approx 505,000 \text{ hours}
\]

### 3. **Reliability (R(t)):**
Reliability is the probability that an item will perform without failure over a specific period. It is calculated as:

\[
R(t) = e^{-\lambda t}
\]

Where \( t \) is the time period for which you want to calculate the reliability.

For example, to calculate the reliability over 1000 hours:

\[
R(1000) = e^{-1.98 \times 10^{-6} \times 1000} \approx e^{-0.00198} \approx 0.99802
\]

This means there is approximately a 99.8% chance that an item will not fail within 1000 hours.

### Summary of Results:
- **Failure Rate (λ):** \( \approx 1.98 \times 10^{-6} \text{ failures per hour} \)
- **MTBF:** \( \approx 505,000 \text{ hours} \)
- **Reliability for 1000 hours (R(1000)):** \( \approx 0.99802 \)

### Do you want more details or have any questions?

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Here are some related questions:
1. How does the failure rate change if the test period is doubled?
2. What happens to the reliability if the number of failures increases to 20?
3. How do you calculate the reliability for a different time period, say 2000 hours?
4. What is the impact of reducing the number of items tested on the reliability estimate?
5. How is the MTBF interpreted in practical applications?

**Tip:** The reliability of a system typically decreases over time, so it's crucial to know the period you're calculating reliability for.

Learn how to calculate failure rate, mean time between failures (MTBF), and reliability for a given test period. Understand the impact of failures on reliability over time. Detailed solutions and formulas provided.

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