Question

For property crime (burglary, car theft, etc) in a particular region, around 70% of the crimes are not solved within 30 days.

a) For 6 property crime cases under investigation, what is the probability that:

i) all of the cases will remain unsolved?

ii) exactly 4 cases will be solved?

iii) at least 4 cases will be solved?

b) For 40 property crime cases under investigation,

i) how many would you expect to be solved within 30 days?

ii) what is the probability that more than 15 of the cases will be solved within 30 days?

Answer 1

70% of the crimes are not solved in 30 days        
p = 0.70        
         
a) n = 6 Sample size under investigation       
Let X be the number of cases out of 6 which remain unsolved in 30 days        
Then X follows a Binomial Distribution with parameters        
n = 6 and p = 0.7        
PDF of X is given by        
         
         
i) To find P(X=6)        
We use the Excel function BINOM.DIST to find the probability        
P(X = 6) = BINOM.DIST(6, 6, 0.7, FALSE)    (Since X = 6 is to be calculated, last parameter is FALSE)    
                = 0.1176        
P(all cases remain unsolved) = 0.1176        
         
ii) To find P(Exactly 4 cases will be solved)        
that is to find P(exactly 2 cases will be unsolved)        
that is to find P(X = 2)        
We use the Excel function BINOM.DIST to find the probability        
P(X = 2) = BINOM.DIST(2, 6, 0.7, FALSE)    (Since X = 2 is to be calculated, last parameter is FALSE)    
                = 0.0595        
P(Exactly 4 cases will be solved) = 0.0595       
         
iii)To find P(atleast 4 cases will be solved)        
that is to find P(atmost 2 cases will be unsolved)        
that is to find P(X ≤ 2)        
P(X ≤ 2) = BINOM.DIST(2, 6, 0.7, TRUE)    (Since ≤ is to be calculated, last parameter is TRUE)    
                 = 0.0705        
P(atleast 4 cases will be solved) = 0.0705        
         
b) n = 40 Sample size under investigation       
Let Y be the number of cases out of 40 which are solved in 30 days        
70% crimes are not solved        
=> 30% crimes are solved in 30 days        
Then Y follows a Binomial Distribution with parameters        
n = 40 and p = 0.3        
PDF of Y is given by        
        
         
i) For a Binomial distribution Expected value = n*p        
Expected value of Y = 40 * 0.3        
                  = 12       
Expected number of cases that would be solved in 30 days = 12        
         
ii) To find P(Y > 15)        
P(Y > 15) = 1 - P(Y ≤ 15)        
                   = 1 - BINOM.DIST(15, 40, 0.3, TRUE)        
                   = 1 - 0.8849        
                   = 0.1151        
P(more than 15 cases are solved in 30 days) = 0.1151