Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 8 marks) 8 3. A B C Figure 1 There are three lily pads on a pond. A frog hops repeatedly from one lily pad to another. The frog starts on lily pad A, as shown in Figure 1. In a model, the frog hops from its position on one lily pad to either of the other two lily pads with equal probability. Let pn be the probability that the frog is on lily pad A after n hops. (a) Explain, with reference to the model, why p1 = 0 (1) The probability pn satisfies the recurrence relation pn +1 = 1 2 (1 –  pn )    n  1  where p1 = 0 (b) Prove by induction that, for n  1 pn = 2 3 1 2 n + 1 3 (6) (c) Use the result in part (b) to explain why, in the long term, the probability that the frog is on lily pad A is 1 3 (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________