Answer to Question #239527 in Statistics and Probability for Cutewow

Question

Canning Town is a blue-collar borough in London, where the borough council tries to squeeze every penny out of the residents. The council set up cameras to observe the parking images in the streets. The records show that 15% of vehicle owners park without pay.

a.What is the likelihood that 3 or 4 vehicles are parked without pay if a random sample of 10 parking images are selected

b.Calculate the probability that no less than 2 vehicles are parked without pay if 20 parking images are randomly selected.

c. What is the mean (average) number of vehicles that are parked without pay, when one hundred parking images are randomly selected

3.2 The N3 highway is one of the important links between Durban and Johannesburg. A road maintenance company found that there are, on average, 8 potholes per every 10km.

a.What is the probability that there are 48 potholes per given 60km

b.What is the standard deviation of the number of potholes in the total length of N3 that is around 570km

Answer 1

1.a.

Binomial distribution:

$P(x=k)=C^k_np^kq^{n-k}$

$p=0.15,\ q=1-p=0.85$

$n=10$

$P(3\ or\ 4)=P(3)+P(4)=C^3_{10}\cdot0.15^3\cdot 0.85^{7}+C^4_{10}\cdot0.15^4\cdot 0.85^{6}=$

$=\frac{10!}{3!7!}\cdot 1.08\cdot 10^{-3}+\frac{10!}{4!6!}\cdot 1.91\cdot 10^{-4}=0.1296+0.0401=0.1697$

b.

$n=20$

$P(x\ge 2)=1-P(1)-P(2)=$

$=1-C^1_{20}\cdot0.15\cdot 0.85^{19}-C^2_{20}\cdot0.15^2\cdot0.85^{18}=$

$=1-0.1368-0.2293=0.6339$

c.

$E(X)=np=100\cdot 0.15=15$

2.a.

Poisson distribution:

$P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}$

$\lambda=8,\ k=8$

$P(48\ potholes\ per\ 60km)=P(8\ potholes\ per\ 10km)=\frac{8^8e^{-8}}{8!}=0.1396$

b.

$\sigma=\sqrt{Var(X)}=\sqrt{E(X)}=\sqrt{\lambda}=\sqrt{8}=2.83$