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I have a question about probability. I posted the question below. I'm to the point where I know/can tell I need to use the Bayes Theorem to solve the problem, but I'm not really sure how to implement it.

Basically, I imagine I need to consider the nth jump as independent of if the dolphin jumped the first time. For example, in -a- I know that the probability the dolphin jumps on the first try is going to be adding two equations together (theorem of counting) because the dolphin will jump sometimes when it isn't asked, and sometimes when it is asked. So, in going out the probability tree I know I need to consider the case where it didn't jump at first and it did jump at first, and follow those out to the probability that the dolphin jumped on both. But at the same time, I get confused because I think about the fact that we're just going to check the probability the dolphin jumped given it was given a mackerel, cod, or none all together which is independent of if the dolphin jumped the first time.

I don't know, I think the problem has too much information.

Question: A dolphin trainer offers a dolphin either Mackerel or Cod as a reward for jumping when asked to. He knows the dolphin prefers Mackerel over Cod. He gives the dolphin Mackerel 4 out of 5 times when she jumps and every other time when she doesn’t jump when asked to. The dolphin only jumps when asked to the next time if given a Mackerel on a try. Let p be the probability that the dolphin jumps on the first try. Assume the dolphin has an infinite appetite and set up a difference equation for the probability that the dolphin jumps the nth time she is asked to.

-a- What is the probability that the dolphin jumps on the nth try?

-b- What is the probability that the dolphin gets a Mackerel on the nth try?

-c- Find the limit as n → ∞ for the answers in (a) and (b).

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You are given

The probability of being given mackerel when jumping: $\newcommand{\P}{\operatorname{\sf P}} \P(M_k\mid J_k)=\tfrac 45$

That of being given mackerel when refusing to jump: $\P(M_k\mid J_k^\complement) = \tfrac 12$

The fact that the dolphin will jump next time only when given mackerel this request, $J_{k+1} = M_{k}$

So $$\begin{align}\P(J_{n}) & = \P(J_{n}\mid J_{n-1})\P(J_{n-1})+\P(J_{n}\mid J_{n-1})\P(J_{n-1}^\complement) \\ & = \tfrac 45\P(J_{n-1})+\tfrac 12\P(J_{n-1}^\complement)\end{align}$$

With the special case of $\P(J_1)=p$

This leads to a recurrence formulation.   Resolve the recursion.

$$\begin{align}p_1&=p\\\ p_{n} &= \tfrac 45 p_{n-1}+\tfrac 12(1-p_{n-1})\\ & = \tfrac 12+\tfrac 3{10}p_{n-1} \end{align}$$

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    "The fact that the dolphin will jump next time only when given mackerel" would make this problem easy, but the fact is that the dolphin will jump with or without the mackerel - it is just more likely given a mackerel was administered. So, that's where I get confused. – 2017-02-17
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    It said: " The dolphin only jumps when asked to the next time if given a Mackerel on a try. " – 2017-02-17
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    I get confused in the wording there, because it sounds like they're saying that the Mackerel affects if the dolphin jumps when asked but not if the dolphin jumps without being asked (which the problem says the dolphin does) – 2017-02-17
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    No, the trainer *gives out* mackerel depending on whether the dolphin jumps on request: 4 in 5 times when it does, and 1 in 2 times when it does not. $$\text{He gives the dolphin Mackerel ((4 out of 5 times) when (she jumps) and (every other time) when (she doesn’t jump)) when (asked to). }$$ Well, okay, the nesting of phrases in that sentence *could* have been made clearer. Does that help? – 2017-02-17