Question

NAME: 7. (15 points.) Left-handed people are more prone to accident-related injury than right-handed people Among a certain population of college students, the number of injuries X reported by a given student is a Poisson random variable; left-handers (L) report injuries at a handers (R) at a rate of 0.15 per year, 15% of the students are left-handed. rate of 0.25 per year, and right- (a) What is the probability that a randomly selected student is injured exactly twice this year? (b) Suppose George, a student in this population, is injured twice this year. Calculate the probability that George is left-handed.

Answer 1

7)

Let X and Y denote the number of injuries for left handers and right hander respectively.

X ~ Poi(0.25), y ~ Poi(0.15)

P(X = r) _ 0.25%-0.25 , P(Y = ,,) _ 0.15%-0.15 r!

x=0,1,2,3,........... and y=0,1,2,3,..........

a)

Required probability = P(student is injured exactly twice)

= P(student is injured exactly twice and student is left handed) + P(student is injured exactly twice and student is right handed)

= P(student is injured exactly twice | student is left handed) . P(student is left handed) + P(student is injured exactly twice | student is right handed) . P( student is right handed)

=

0.25"e-0.25 21ㅡㅡㅡ × 0.15 + 0.15%-0.15 × (1-0.15) = 0.0119

b)

Let E1 and E2 denote the event that person is left hander and right hander respectively.

Let A denote the event that the person is injured twice per year.

Using Bayes theorem,

Required probability =

P(E1)P(AE1)

0.15P(X = 2) 2) + (1-0.15)P(y = 2) 0.15P(X

0.15 × 0.25t- 2! 0.15 × 0.252e- 35 ,-ー 0.152-TE = 0.3073 0.85