A botanical research station monitors two rare plant species, S1 S_1\,S1 and S2S_2S2, for signs of a specific fungal blight. Let A A\,A be the event that species S1 S_1\,S1 is infected and B B\,B be the event that species S2 S_2\,S2 is infected. The probabilities of these events are given by
P(A)=x,P(B)=y,P(A∪B)=0.6,P(B∣A)=0.4 P(A) = x, \quad P(B) = y, \quad P(A \cup B) = 0.6, \quad P(B|A) = 0.4 P(A)=x,P(B)=y,P(A∪B)=0.6,P(B∣A)=0.4Show that
3x+5y=3 3x + 5y = 3 3x+5y=3The station also tracks a second pathogen. Let C C\,C be the event that species S2 S_2\,S2 is infected with this second pathogen. It is known that B B\,B and C C\,C are mutually exclusive such that
P(B∪C)=0.65,P(C)=0.1x+y P(B \cup C) = 0.65, \quad P(C) = 0.1x + y P(B∪C)=0.65,P(C)=0.1x+y(i) Find a second equation in x x\,x and y y\,y that does not involve fractions.
(ii) Hence find the value of x x\,x and the value of yyy.
Determine whether or not A A\,A and B B\,B are statistically independent. You must show your working clearly.
