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)=0.4P(A) = x, \quad P(B) = y, \quad P(A \cup B) = 0.6, \quad P(B|A) = 0.4P(A)=x,P(B)=y,P(A∪B)=0.6,P(B∣A)=0.4
Show that
3x+5y=33x + 5y = 33x+5y=3
The 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+yP(B \cup C) = 0.65, \quad P(C) = 0.1x + yP(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.
Practise Edexcel A Level Maths Conditional Probability with exam-style questions for A Level Maths. 100 questions covering Set Notation, Conditional Probability, Conditional Probabilities in Venn Diagrams, Probability Formulae, and Tree Diagrams, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.