According to a national survey, in 80% of property crimes, the criminals are nev

Question

According to a national survey, in 80% of property crimes, the criminals are never found and the cases remain unsolved. Supposed in a certain neighborhood district, the police are investigating six property crimes. How many property crimes "n" must the police investigate before they can be at least 90% sure of solving one or more cases?

All I can come up with is:
P(0.80) = probability of unsolved crimes out of the total crimes in the district, and
P(r = 1) = 0.90 = probability of at least one crime before police are 90% sure

Any suggestions or ideas to help out?

Explanation / Answer

This follows a negative binomial distribution. Mean is r/p where mean is the average number of trials to produce r successes and p is the probability of success. Here r=1 and p=0.2. Variance is r(1-p)/p^2 so standard deviation is the square root of this. 1/.2= 5 (mean) sqrt r(1-p)/p^2= 4.47 is the standard deviation. A z score of 1.28 (from a normal table) provides .90 probability. (x-5)/4.47= 1.28 so x= 10.72= 11 investigations.

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