Question

Statisticssssss -_______-

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 three independent hard disk drives, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Round to four decimal places.

Answer 1

This is a binomial distribution.

Answer 2

\[
X\sim B(n,p) \implies \Pr(X=k) = \binom nk p^{k}(1-p)^{n-k}
\]

Answer 3

Yeah, I can.

Answer 4

Do you know what \(\binom n k\) is?

Answer 5

n choose k

Answer 6

Ok. So if you have two drives, how many ways are there for one to remain?

Answer 7

Yes

Answer 8

What is the probability for 1 to remain?

Answer 9

Not quite.

Answer 10

Okay suppose we give them a letter. There is dive A and drive B.

Answer 11

What is the probability that drive A fails and the probability that drive B doesn't fail?

Answer 12

"Assume that there is a %8 rate of disk drive failure in a year."

Answer 13

What is the probability that drive A fails?

Answer 14

That is the simplest I can get it.

Answer 15

92% is that it doesn't fail.

Answer 16

Now, does the failure of drive A affect the failure of drive B? Is it independent?

Answer 17

Well it is independent.

Answer 18

They are independent because they don't affect each other. Drive A will fail or not fail with the same probability regardless of whether drive B fails.

Answer 19

What is the probability of two independent events happening? Do you remember any formula?

Answer 20

This isn't conditional probability.

Answer 21

What is the probability of BOTH independent events happening.

Answer 22

\[
\Pr(AB) = \Pr(A) \times \Pr(B)
\]

Answer 23

When \(A,B\) are independent.

Answer 24

IN this case \(\Pr(A)= 0.08\) and \(\Pr(B) = 0.92 = 1-0.08\)

Answer 25

And \(A\) is the event drive \(A\) fails and \(B\) is the event \(B\) doesn't fail.

Answer 26

We have 0.0736 that A fails and B doesn't fail.

Answer 27

Next we need the probability that A doesn't fail and B fails.

Answer 28

Let's keep it simple for now.

Answer 29

Okay

Answer 30

What is the probability that A doesn't fail and B fails?

Answer 31

probability that A doesn't fail is .92

Answer 32

probability that B does is .08

Answer 33

yeah but for both?

Answer 34

IT's the same as before

Answer 35

Yes.

Answer 36

Now one last one we need to do.
What is the probability that A doesn't fail and B doesn't fail.

Answer 37

Okay

Answer 38

So we have probabilities for:
A fails, B doesn't
A doesn't, B fails
A doesn't, B doesn't

Answer 39

There are the 3 ways in which at least one drive doesn't fail.

Answer 40

They are mutually exclusive events, so the probability of either of them happening is just the sum of them all

Answer 41

So if we add up all of the probabilities we have calculated we will have the probability that at least one doesn't fail.

Answer 42

That is the answer to a)

Answer 43

No. 0.0736 was just the probability that A fails and B doesn't

Answer 44

The only way to do it faster is to understand it.

Answer 45

There isn't one simple formula for every problem.

Answer 46

Where would you start?

Answer 47

There are a couple formula you should know by now about probability.

Answer 48

Okay good luck.