Answer to Question #338397 in Statistics and Probability for Developer

Question

The probability that a car is stolen when it is parked overnight in an unsafe

area of a city is 10%. If there are 12 cars parked on a particular street of that

area, what is the probability that during a night

(i) no car is stolen?

(ii) at most two cars are stolen?

(iii) at least nine cars are stolen?

Answer 1

Let $X=$ the number of cars stolen:$X\sim Bin(n, p).$

Given $n=12, p=0.1, q=1-p=0.9.$

(i)

$$P(X=0)=\dbinom{12}{0}(0.1)^0(0.9)^{12-0}$$

$$=0.28242953648$$

(ii)

$$P(X\le2)=P(X=0)+P(X=1)+P(X=2)$$

$$=\dbinom{12}{0}(0.1)^0(0.9)^{12-0}+\dbinom{12}{1}(0.1)^1(0.9)^{12-1}$$

$$+\dbinom{12}{2}(0.1)^2(0.9)^{12-2}=0.88913002226$$

(iii)

$$P(X\ge9)=P(X=9)+P(X=10)$$

$$+P(X=11)+P(X=12)=\dbinom{12}{9}(0.1)^{9}(0.9)^{12-9}$$

$$+\dbinom{12}{10}(0.1)^{10}(0.9)^{12-10}+\dbinom{12}{11}(0.1)^{11}(0.9)^{12-11}$$

$$+\dbinom{12}{12}(0.1)^{12}(0.9)^{12-12}< 0.000001$$