Equipment reliability formula question ("A piece of equipment that produces widgets runs for 1000 hours....")

[MontyJ]

Hello all,
I do not know if this is the correct place for this, so please excuse me if it's not.
Here is my problem. It concerns the reliability of a certain piece of machinery. The reliability formula is:

Rt=e^(-xt)

Where R is reliability
t is time
x is failure rate (it should be lambda, but I don't know how to produce a lambda on the keyboard)
e is the base of the natural logarithm (2.71828)
So R(time) = 2.71828^-(lambda x time)

A piece of equipment that produces widgets runs for 1000 hours. During that 1000 hours it breaks down 10 times.

This gives the equipment a MTBF (mean time between failure) of 100 hours.

The failure rate is the inverse of the MTBF or .01

If I want to calculate reliability for 20 hours I get:

R(20) = 2.71828^-(.01 x 20) so R(20) = .8187 or 81.87%

R(100) = 2.71828^-(.01 x 100) so R(100) = .3679 or 36.79%

This makes perfect sense in that the longer it runs, the less reliable it becomes.

However, and here is my problem, if I do the calculation for the original 1000 hours, I get:

R(1000) = 2.71828^-(.01 x 1000) so R(1000) = 4.54 or 454%

I'm screwing this decimal up somewhere but high school math was over 40 years ago. Can someone please explain what I'm doing wrong here?

Thanks for your time.

 

[BigBeachBanana]

Wolfram Alpha (Online calculator) tells me that...

 

[Dr.Peterson]

However, and here is my problem, if I do the calculation for the original 1000 hours, I get:

R(1000) = 2.71828^-(.01 x 1000) so R(1000) = 4.54 or 454%

I'm screwing this decimal up somewhere but high school math was over 40 years ago. Can someone please explain what I'm doing wrong here?

I suspect your calculator gave an answer like

4.5399929762484851535591515560551e-5​

and you either didn't see the "e-5", or didn't know that it means "[imath]\times10^{-5}[/imath]".

 

[Steven G]

It's easier to type out e compared to 2.71828, so why do you use 2.71828 instead of e--they are even equal to one another!

 

[Dr.Peterson]

It's easier to type out e compared to 2.71828, so why do you use 2.71828 instead of e--they are even equal to one another!

No they aren't. I thought you were a stickler for not saying two things are equal when one is an approximation of the other!

But I suspect the OP is not aware there's an e (or e^x) button on the calculator. That's not uncommon.

 

[Steven G]

Darn, another typo! I'm sure that you knew I meant to write they are not even equal to one another!