The Elsmar Cove Forum Probability problem - The probability of picking 4 bad brakes out of 500
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#1
16th January 2006, 10:22 AM
 anoop_mahajan Anoop   Registration Date: Feb 2005 Location: Michigan Posts: 15 Thanks Given to Others: 0 Thanked 0 Times in 0 Posts Karma Power: 35 Karma: 20
Probability problem - The probability of picking 4 bad brakes out of 500

We know that there are 15 bad brakes in a lot of 500. We need to pick 4 brakes for use.
The probability of picking 1 bad brake = 15/500 = 3%
The probability of picking 4 bad brakes = 15/500+14/499+13/498+12/497 = 10.83 %

Am I right ?

#2
16th January 2006, 10:27 AM
 roland_lu Involved - Posts   Registration Date: Jul 2005 Location: Canada Posts: 95 Thanks Given to Others: 2 Thanked 1 Time in 1 Post Karma Power: 41 Karma: 135
15/500 *14/499*13/498*12/497= very unlikely the 4 u picked are all bad
#3
16th January 2006, 11:22 AM
 Tim Folkerts Forum Moderator   Registration Date: Sep 2003 Location: Kansas, USA Age: 50 Posts: 975 Thanks Given to Others: 29 Thanked 351 Times in 211 Posts Karma Power: 148 Karma: 5804
The probability of picking a single bad brake is indeed

15/500 = 3%.

However, the probability of picking a bad brake followed by 3 good brakes is

15/500 * 485/499 * 484/498 * 483/497 = 2.75%

But the bad one could be drawn 1st, 2nd,3rd or 4th, so multiply by 4 -> 11.0% that one of the four is bad!

You can come mighty close just using the binomial distribution. In Excel that would be =BINOMDIST(1,4,0.03,FALSE) which gives 10.9% Binomial assumes that there is a 3% chance that the next is bad, no matter what the ones before were. Since the lot is large (500) and the sample is small (4), this approximation should worke pretty well.

I agree with roland_lu for the odds of all four being bad.
Tim F
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#4
16th January 2006, 01:24 PM
 Steve Prevette Deming Disciple   Registration Date: Feb 2004 Location: Aiken, SC Posts: 1,996 Thanks Given to Others: 140 Thanked 1,053 Times in 558 Posts Karma Power: 259 Karma: 11184
If you want to come somewhat closer than the binomial, use the hyper geometric. The hypergeometric does not cumulate in Excel, so you have to write out the probabilities for 0, 1, 2, 3, and 4 bad brakes. Or simply subtract the probability of no bad brakes from 1.

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