Assume that there is a 8​% rate of disk drive failure in a year.
a. If all your computer data is stored on a hard disk drive with a copy stored on a second hard disk​ drive, what is the probability that during a​ year, you can avoid catastrophe with at least one working​ drive?
b. If copies of all your computer data are stored on four independent hard disk​ drives, what is the probability that during a​ year, you can avoid catastrophe with at least one working​ drive?
Question content area bottom
Part 1
a. With two hard disk​ drives, the probability that catastrophe can be avoided is 
  
0.9936.
​(Round to four decimal places as​ needed.)
Part 2
b. With four hard disk​ drives, the probability that catastrophe can be avoided is 
  
enter your response here.
​(Round to six decimal places as​ needed.)

Let's solve both parts of the problem step-by-step.

### Part (a): Two Hard Disk Drives

1. **Given Information:**
   - Probability of failure for one hard disk drive in a year = 8% = 0.08.
   - Probability of success (working) for one hard disk drive = 1 - 0.08 = 0.92.

2. **Calculating the Probability of Avoiding Catastrophe:**
   - For catastrophe to occur, both drives must fail.
   - Probability that both drives fail = \( 0.08 \times 0.08 = 0.0064 \).

   - Probability that at least one drive works (avoiding catastrophe) = \( 1 - 0.0064 = 0.9936 \).

The answer for part (a) is:
\[ \text{Probability} = 0.9936 \]

### Part (b): Four Hard Disk Drives

1. **Calculating the Probability of Avoiding Catastrophe:**
   - Probability of all four drives failing = \( (0.08)^4 \).

   Let's compute this:

\[
(0.08)^4 = 0.00004096
\]

2. **Probability of at Least One Drive Working:**
   - Probability of at least one drive working = \( 1 - (0.08)^4 = 1 - 0.00004096 \).

\[
1 - 0.00004096 = 0.99995904
\]

Rounding to six decimal places:

\[ \text{Probability} = 0.999959 \]

So, the answer for part (b) is:
\[ \text{Probability} = 0.999959 \]

Would you like more details, or do you have any other questions? Here are some related questions you might find interesting:

1. What would be the probability of avoiding catastrophe with three independent hard drives?
2. How does the probability change if the failure rate increases to 10% per year?
3. What is the probability that at least two of the four hard drives fail?
4. How many hard drives would be needed to achieve a 99.999% reliability rate?
5. How would correlated failures between drives impact the overall probability?

**Tip:** When dealing with reliability problems, redundancy (using multiple independent backups) significantly increases the probability of avoiding data loss.

Assume that there is an 8% rate of disk drive failure in a year. a. If all your computer data is stored on a hard disk drive with a copy stored on a second hard disk​ drive, what is the probability that during a​ year, you can avoid catastrophe with at least one working​ drive? b. If copies of all your computer data are stored on four independent hard disk​ drives, what is the probability that during a​ year, you can avoid catastrophe with at least one working​ drive?

Explore how to calculate the probability of avoiding data loss with multiple hard disk drives. Learn the step-by-step process for determining reliability with two and four independent hard drives, applying key probability formulas. Suitable for advanced high school and college students.

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