Wikipedia:Reference desk/Archives/Mathematics/2007 November 16

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November 16

Probability of multiple drive failure

Say you've got 4 hard drives in a RAID, each with a probability of .2 of failing in a certain amount of time. What are the odds of at least two of them failing within that amount of time?

My first guess is that the probability of 2 out of 2 drives failing is .2 x .2 or .04, but for 2 out of 4 drives failing, there are 6 failure events (drives 1 & 2 failing, 1 & 3, 1 & 4, 2 & 3, 2 & 4, 3 & 4) each with a probability of .04, so the answer must be 1 - (1 - .04) ^ 6 or .217 . However, I think this might be wrong because the 6 failure events all overlap each other, i.e. aren't mutually exclusive.

--Clairvoyant walrus 02:35, 16 November 2007 (UTC)

The probability that none fail is 0.8^4, and the probably that only the first drive fails is 0.2*0.8^3, only the second is 0.8*0.2*0.8^2, etc. for a total probability of 0.8^4 + 4*0.2*0.8^3 = 0.8192 that at most one drive fails. Switching back to the original, the probability is 1-0.8192=0.1808 that at least two drives fail during the time period. JackSchmidt 02:43, 16 November 2007 (UTC)

(edit conflict)

Think about this, how many combinations can two six-sided die make? 36 which is also $6^{2}$ and we had two dice, for three it's $216=6^{3}$ So we know fourth powers are involved with you drives. Since your probability is 1/5 in fraction form, we can thing of your drives as five-sided die. So you basically asking what's the chance that two or more of your four dice will roll a five. It is easier to subtract the probability that one or no drives will fail from 1 to get your answer. P(none fail)=P(no 5) is easy to find, We know that the chance that any one drive doesn't fail is 4/5 so $P(no5)=(4/5)^{4}$. Finding the Probability that one fails (P(one fails)) is more difficult, it involves a binomial expansion. We know that one of drives must fail so it's chance of happening is 1/5 and we know that none of the others can fail so that's (4/5)^3, and since there are 4 distinguishable ways one drive can fail we must multiply by 4. So we get 4 times P(one fails) times P(three don't fail)= 4(1/5)(4/5)^3=(4/5)^4 and add that two (4/5)^4 and subtract from one. That should do it. A math-wiki 03:19, 16 November 2007 (UTC)

To add to the above, if P = one single specific disk failing and Q = that disk not failing, two complementary events, and the events for separate disks are independent, then the possibilities for 4 disks are summed up symbolically by (P+Q)×(P+Q)×(P+Q)×(P+Q). If the identity of the disks is immaterial, QP = PQ, and by multiplying out the product and collecting terms with equal powers of P, you get all the ways you can get n simultaneous disk failures, for n from 0 to 4. The terms that are relevant here are those with powers at least 2: 6P²Q² + 4P³Q + P⁴. Now replace Q by 1−P, put P = 0.2, calculate, and there you are. See Binomial distribution for the general case.  --Lambiam 08:50, 16 November 2007 (UTC)

There are ⁴C₂ = 6 possible combinations of drives to fail. Each individual drive has a 0.2 possibility of failing. Each combination therefore has 0.2 * 0.2 = 0.04 possibility of occuring. There are 6 possible combinations, so 6 * 0.04 = 0.24 which is the probability of precicely two drives having failed.

If you repeat for three drives and four drives and add those results together you get the total for at least two drives failed.

Tom-hundered-% 13:45, 16 November 2007 (UTC)

Not quite. The probability of exactly two drives failing is 6 * 0.2 * 0.2 * 0.8 * 0.8 = 0.1536 (not 0.24); the probability of exactly three drives failing is 4 * 0.2 * 0.2 * 0.2 * 0.8 = 0.0256; and the probability of all four drives failing is 0.2 * 0.2 * 0.2 * 0.2 = 0.0016. Add these three cases together (noting that they are mutually exclusive) and you find that the probability of two or more drives failing is 0.1808. Same answer as JackSchmidt, but following Lambian's route. Gandalf61 14:18, 16 November 2007 (UTC)

Gandalf61, you're right. I guess my probability is a bit rusty :| Tom-hundered-% 14:35, 16 November 2007 (UTC)

Tom, you started in the right way; you found the number of combinations for the case of two (out of four) disks failing. The point is that ⁴C₂ must be then multiplied by the probability (out of combinations) of two disks out of four going down. That last term is precisely 0.2 * 0.2 * 0.8 * 0.8 = 0.1536, as Gandalf61 points out. You just made ⁴C₂ times the probability of the pair of disks failing, but did not account for the fact that the remaining two must not fail, which is given by the last two factors [0.8 * 0.8] Pallida Mors 22:11, 16 November 2007 (UTC)

limit

Excuse me, but are left hand limits and normal limits the same thing? ---- 142.132.6.69 (talk) 20:16, 16 November 2007 (UTC)

No, they are not. If you have a function f(x) that is 1 if x is less than or equal to 0, and 0 if x is greater than 0, then the limit at that point does not exist, but the left limit is 1 and the right limit is 0. A sided limit is the limit when you're going from one side or the other to point, and a a normal limit exists only when the left and right limit exist, and are equal. See Limit_of_a_function#Definitions -- Gscshoyru (talk) 20:20, 16 November 2007 (UTC)

The traditional simple case is to consider $f(x)=\left\{{\begin{array}{ll}0&x<0\\1&x>0\end{array}}\right.$. We can compare $\lim _{x\to 0^{+}}f(x)=1$, $\lim _{x\to 0^{-}}f(x)=0$ and $\lim _{x\to 0}f(x)$ which is undefined. -- Donald Hosek (talk) 22:36, 16 November 2007 (UTC)

Fixed that for you. –King Bee ^{(τ • γ)} 15:45, 17 November 2007 (UTC)

Opps. Um... Indeterminate and undefined are two distinct diffrent things. Because the left hand limit cannot equal the right hand limit, the normal limit is indeterminiate. ( It cannot be determined weather its 0 or 1. ) Much more on this will be discussed on user Talk A_Math, as soon as I find his page.

Huh? "Undefined" means "not defined", and usually refers to a case where a standard definition does not apply. "The limit" of a function is only defined for points where it exists, and no limit exists for f at 0, thus "the limit of f at 0" is undefined. "Indeterminate" is a special case - it refers to some arithmetic operation which is not defined in the standard sense and the definition cannot be extended to it by considering limits. For example, 0/0 is indeterminate, because knowing only that two functions tend to 0, we can't deduce what the limit of their ratio is. -- Meni Rosenfeld (talk) 18:01, 23 November 2007 (UTC)