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 perpetuated 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

Step-by-step explanation:

Given that 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†.

WE find that each crime is independent of the other and hence the probability out of five property criminal cases in this district not to be solved is binomial with constant prob = 0.80

X is binomial (n = 5, p =0.80)

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

P(x=5) = 0.32768

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

P(atleast one crime is solved ) = 1-P(no crime is solved)

= 1-P(X=0) = 1-0.0032 = 0.99968

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

Crimes that will be solved has probability as 0.20

E(X') = 5(0.2) = 1

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?

Given that P(X=0) = 1-0.90 = 0.10

So n should be between 10 and 11

n = crimes