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PERT - need assistance with probability calculation


Please find attached XLS that calculates the probability (using PERT). I am not entirely clear though between the "within" and "at least" calculation.

For right now, my calculations are based on "within".  How should I modify the calculations to determine the probability for the "at least"?

Thank you,
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P(at least n months) = 1-P(within n months)
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Perfect solution!!!
d-glitch, ozo:

I may have prematurely closed the posting.

Please see attached snapshot.   I am still not 100% confident that the "at least" is  being accounted for.  

Maybe ozo's recommendation of P(at least n months) = 1-P(within n months) was correct.   Currently, d-glitch's response is accurate when determining "within"... not sure about the "least".  

I would appreciate if you provide me your answers for the two "at least" probabilities.

Thank you... I'm just a bit rusty on calculating these.

... forgot to attach the snapshot.
At least 17 months  means  17 months or more  ==>   1  -  Prob(16 months or less)

You can also use Excel for the latest problem as well.  
The norm.dist() function is the most useful by far, because they have done the integration.
You need to know the mean and the standard deviation (which is sqrt(81)=9 in this case).

in 52 weeks   means   prob( 52 weeks or less)  -  prob( 51 weeks or less)

longer than 65 weeks   means  prob( 1  -  65 weeks or less)

Thanks for the feedback...  I am still not entirely clear on the following:

a) require at least 17 months?
c) require at least 23 months?

Based on your proposed method (see attached XLS), could you please help me -- in XLS -- define those calculations.

I am inclined to use a VLookup based on the value selection ("at least" or "within) in cell range E8:E11.

I am sure I can tweak this... at this moment, I nearly need to know what the actual probabilities are for the four selected values (17, 20, 23, 25).

Any additional help would be greatly appreciated.

I think you should use the norm.dist formula rather than a lookup table.
The table doesn't help if you have fractions of a month, but the formula works just fine.

The cumulative version of the formula gives you the within probability directly.
     within 20 months = 0.3415
     within 25 months = 0.9488

At least 17 months means exactly the same thing as not within 16 months.
     at least 17 months  ==>  not within 16 months  =  (1 - 0.0206)
     at least 23 months  ==>  not within 22 months  =  (1 - 0.6568)

If you specify a single month . . .    What is the probability that it will take 22 months
Then you probably mean more than 21 months, but not more than 22 months.
    prob(22) - prob(21)  =  (0.6585 - 0.5000)

Thank you... that helped a lot.  

I've used your solution to determine the probabilities via a Lookup (from the Norm.dist) table.

Based on the numeric values AND dropdown values (cell range E8:E12), the correct probabilities are determined.  See attached XLS as the solution.

Again, thank you for your help.

> At least 17 months means exactly the same thing as not within 16 months.
16.5 months is neither "at least 17 months", nor "within 16 months".

Thanks for chiming in... are you suggesting the latest posted XLS includes incorrect elements?

ozo is certainly correct in the mathematical sense, and it was incorrect for me to make assumptions without stating them explicitly.

I did notice that the specification of the problem relied on integers, and that you were using the NORM.DIST function in Excel which uses real numbers.  
The probability of any real-world mean of a real-number process being 12.000... is very small (zero).

Integers are often appropriate in legal and commercial transactions.  
You can eat 16.5 apples, but you probably have to purchase 17.
The cost of airport parking is $20 per day or any fraction there of.
This is the method I was assuming.
From your second question:
   >>   Compute the probability the project will be completed in 52 weeks.

What does 52 weeks mean?  How are you treating numbers?  How do you want to treat numbers?  

The probability that the project will be completed in exactly 52.000... weeks is zero.

The probability that it will be completed at           t  such that   51  <    t  <= 52
and the probability that it will be completed at    t  such that   51.5  < t  <= 52.5
are close but not equal.

Thank you for the follow-up and the additional information.

At this moment, I am mostly interested in "at least" and "within".    Also, the assumption that I'm dealing only with integers is correct.

Based on that, do you agree with the calculation presented in the most recently posted XLS?

Precise math requires precise language as well.  Just because you are get bad language, doesn't mean you have to give it back.  [That is my error on this question.]

The probabilities for within 20 and within 25 are correct, as long as within 20 months means in 20.000... months or less.  The second phrase is more precise.

more than 21, but not more than 22  is a precise answer.

ozo is correct.  at least 17 months remains problematic.
You can make a case for not within 16 months.
But it wouldn't hold up to a decimal point.   at least 17.0 months  has a different meaning.

Interpret the question as best you can, then make a precise answer.