According to a survey 65 of murders committed last year were

According to a survey, 65% of murders committed last year were cleared by arrest or exceptional means. Fifty murders committed last year are randomly selected, and the number cleared by arrest or exceptional means is recorded. (a) Find the probability that exactly 40 of the murders were cleared. (b) Find the probability that between 36 and 38 of the murders, inclusive, were cleared (c) Would it be unusual it fewer than 18 of the murders were cleared? Why or why not? Round to four decimal places.

Solution

probability that the murders are cleared = 0.65

probability murder not cleared = 1 - 0.65 = 0.35

total murders = n = 50

a) P(40) = 50C40 (0.65)^40(0.35)^10 = 0.0093

b) between 36 to 38 = p(36 )+p(37)+p(38)

50C36*(0.65)^36(0.35)^14 + 50C37 (0.65)^37(0.35)^13 + 50C38 (0.65)^38(0.35)^12

= 0.1535

c) P(X < 18) = SUM x from 0 to 17: 50Cx (0.65)^x (0.35)^(50-x)

= 0.00007

The mean = np = 32.5

The standard deviation = sqrt (npq) = sqrt (50*0.65*0.35) = 3.37

Now, 18 is (32.5 - 18) / 3.37 = 4.30

hence as this is less then the mean it will be unusual.