Could you please provide the calculation process, thank you!

[c] [3 marks] Now you are to nd a general formula for p\" in terms of :1. There are different ways to do this, but here 1 require you to use a linear recursion where the variables are: p" = probability even # heads after u ips on = probability odd # heads after u ips. Elf course on = 1 p" but by carrying both along, we can get a 2x2 matrix recursion. Foreiulate and solve this recursion and get a formula for on. Check that your result gives the same value for 3:33 as you obtained in the table. [There are other checks you can do as well. For example you already know the answer for three special cases of h.] Q6. [Eu marlts] Having solved Q5 you have a formula for p\" the probability of getting an even it heads after u flips of a biased coin. Well that sets you up with another way ofsolying the probleminduction. Recall that the trouble with induction is that you need a formula to start with--that's what you use as your inductive hypothesis. And most of the time we don*t have such a formula. But now we do. So in this question 1 want you to take your formula for p\" and establish it using mathematical induction. The interesting thing about this problem is that if you made a mistake in your Formula for p\". the induction wont work! So this seryes as a way for you to check your answer. Q5. (a) [2 marks] Suppose you flip n fair coins. Let p, be the probability that you get an even number of heads. Find pn. You have to justify your answer (give some sort of argument). Now we ask the same question as (a) except that the coins are not fair but come up heads with probability h n Pn and tails with probability 1 - h. 0 (b) [3 marks] In this case, p, will depend on h. By considering the different possibilities, find p, for n 1 between 0 and 3 and put your values in the table at the right. 2