Question

Does crime pay? The FBI Standard Survey of Crimes showed that for about 80% of all property crimes (burglary, larceny, car theft, etc.), the criminals are never found and the case is never solved†. Suppose a neighborhood district in a large city suffers repeated property crimes, not always perpetrated by the same criminals. The police are investigating five property crime cases in this district.

(a) What is the probability that none of the crimes will ever be solved? (Round your answer to three decimal places.)


(b) What is the probability that at least one crime will be solved? (Round your answer to three decimal places.)


(c) What is the expected number of crimes that will be solved? (Round your answer to two decimal places.)
crimes

What is the standard deviation? (Round your answer to two decimal places.)
crimes

(d) How many property crimes n must the police investigate before they can be at least 90% sure of solving one or more cases?
n =  crimes

Answer 1

Answer)

As there are fixed number of trials and probability of each and every trial is same and independent of each other

Here we need to use the binomial formula

P(r) = ncr*(p^r)*(1-p)^n-r

Ncr = n!/(r!*(n-r)!)

N! = N*n-1*n-2*n-3*n-4*n-5........till 1

For example 5! = 5*4*3*2*1

Special case is 0! = 1

P = probability of single trial = 0.8

N = number of trials = 5

R = desired success

A)

P(0) = 5c0*(0.8^0)*(1-0.8)^5-0 = 0.00032

B)

P(at least 1) = 1 - p(0)

But here p = 1 - 0.8 = 0.2 (as probability that case will be solved is = 20%)

0.67232

C)

Expected number of crimes that will be solved = n*p = 5*0.2 = 1

S.d = √{n*p*(1-p)} = 0.89442719099

D)

P = 0.2

P(at least 1) = 1 - p(0)

0.9 = 1 - nc0*(0.2^0)*(0.8)^n

0.8^n = 0.1

Taking log on both sides

Nlog0.8 = log 0.1

N = log 0.1/log 0.8

N = 10