Independence of long times coin toss model

Question

A coin is tossed independently $n$ times. The probability of heads at each toss is $p$. At each time $k (k = 2,3,\cdots,n)$ we get a reward at time $k+1$ if $k^{th}$ toss was a head and the previous toss was a tail. Let $A_k$ be the event that a reward is obtained at time $k$.

a.Are events $A_k$ and $A_{k+1}$ independent? b. Are events $A_k$ and $A_{k+2} independent?

In this problem as the event $A_k$ depends on the previous out comes so I think bit a will be dependant however bit b is independent. Is it the correct way to show.

Answer 1

a) $A_k$ and $A_{k+1}$ are not independent since $P(A_{k+1}|A_k) = 0$ because if $A_k$ occured then in $k-1^{th}$ toss there was a head instead the tail necessary for $A_{k+1}$ to occurs.
b)Yes becouse tere is a gap of two tosses allowing to occurs the event $TH$ before $k+1^{th}$ toss.

Answer 2

If $A_k$ implies head, tails. Then $A_{k+1}$ will need also head, tails. The probability for these events is $1/4$ independence would require $P(A_k$ and $A_{k+1})=P(A_k)P(A_{k+1})=1/16$. But we know $P(A_k$ and $A_{k+1})=0$ as it is impossible to get both after each other.

In the second case we loose dependence because the toss from $A_k$ is not important.