Question

State Farm Insurance studies show that in Colorado, 45% of the auto insurance claims submitted for property damage were submitted by males under 25 years of age. Suppose 12 property damage claims involving automobiles are selected at random.

(a) Let r be the number of claims made by males under age 25. Make a histogram for the r-distribution probabilities.

(b) What is the probability that three or more claims are made by males under age 25? (Use 3 decimal places.)


(c) What is the expected number of claims made by males under age 25? What is the standard deviation of the r-probability distribution? (Use 2 decimal places.)

μ
σ

Answer 1

Part a)

B :- Binomial Distribution

X ~ B ( n = 12 , P = 0.45 )

P(X = 2) = CHP'Qin-r)


C = n! (n - 1)!?!

X	P(X)
0	0.0008
1	0.0075
2	0.0339
3	0.0923
4	0.17
5	0.2225
6	0.2124
7	0.1489
8	0.0762
9	0.0277
10	0.0068
11	0.001
12	0.0001

Histogram 1 2 3 4 5 6 7 8 9 10 11 12 13

0.15 plo 0 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12

Part b)

P ( X >= 3 ) = 1 - P ( X <= 2 )

P(X = 0) = C12(0.45)º(0.55) (12-0) = 0.0008


P(X = 1) = C120.45)*(0.55) (12-1) = 0.0075


P(X = 2) = 0.120.45)2(0.55) (12-2) = 0.0339

P ( X >= 3 ) = 1 - 0.0422 = 0.958

Part c)

Mean $\mu$ = n * P = ( 12 * 0.45 ) = 5.4
Variance $\sigma^2$ = n * P * Q = ( 12 * 0.45 * 0.55 ) = 2.97
Standard deviation $\sigma$    = (variance) = (2.97) = 1.72