Question

1, (a) Suppose that 60% of a large population know how the word 'accommodation' is spelled. Five people were selected at random from the population and asked i. What is the probability that exactly 2 of the 5 spelled the word correctly? ii. What is the probability that at least one person spelled the word correctly? ii. Given that the first person spelled the word correctly, what is the proba- (17 marks) (b) It is believed that fatalities due to lightening strikes in the UK follow a Poisson i. What is the probability of 4 lightening strike fatalities in the UK in one i. Suppose that there is a lightening strike fatality in the UK today. What to spell the word. bility that exactly 4 of the 5 spelled it correctly? distribution with a mcan of 3 per year. year? is the probability that the next UK lightening strike fatality is less tharn 6 months away? (16 marks)

Answer 1

1)

here this is binomial distribution with paramter n=5 and p=0.6

i)P(X=2)=⁵C₂(0.6)²(0.4)³ =0.2304

ii) P(X>=1)=1-P(X=0)=1-⁵C₀(0.6)⁰(0.4)⁵ =1-0.0102=0.9898

iii)P(X=4|X>=1)=P(X=4)/P(X>=1)=⁵C₄(0.6)⁴(0.4)¹/0.9898 =0.2592/0.9898=0.2619

b)

here parameter $\lambda$ =3

i)P(X=4)=e^-$\lambda$$\lambda$^x/x! =e^-33⁴/4! =0.1680

ii)expected strike in 6 months =3/2 =1.5

P(next strike in less then 6 months)=1-P(no strike in 6 months)=1-e^-1.5*1.5⁰/0! =1-0.2231=0.7769