probability math question

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 eleven 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?

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

Answer 1

80% cases are never solved 20% of cases are solved 11 cases. Independent

binomial formula (probability x cases will be solved in 11 trials with probability of "success" = p = 0.2)

11! x n-x

----- p (1-p)

x! (11-x)!

for problems here n = 11, p not solved = 0.8, p solved = 0.2

a) p(x = 0) = 11! 11 0

---- 0.8 0.2 = 0.086

0! 11!

b) p(at least 1 case wall be solved) = 1 -p(x = 0) = 0.914

c) expected = np = 11 * 0.2 = 2.2 cases solved.

d) standard deviation = sqrt(11*0.2*0.8) = 1.327