Let $X$ be the random variable that counts the number of tosses until we get two successive tosses of the same type. Although you did not say so explicitly, I assume that you want the probability mass function of $X$.
What is the probability that $X>n-1$? We need to have a sequence of tosses of length $n-1$ and of type HTHTHT and so on (alternating heads and tails, starting with head), or THTHTH and so on. The probability that the first type of sequence is followed up to and including the $(n-1)$-th toss is $\frac{1}{2^{n-1}}$. The second type of sequence has the same probability, for a total of $\frac{2}{2^{n-1}}$.
The number $X$ of tosses is equal to $n$ if $X>n-1$ and the last toss matches the next to last toss. The probability of that is $\frac{1}{2}$, so
$$P(X=n) =\frac{2}{2^{n-1}}\cdot\frac{1}{2}.$$
This simplifies to $\dfrac{1}{2^{n-1}}$. Note that $n$ ranges over the integers that are $\ge 2$.
Another way: The first toss doesn't matter. After that, it is just a matter of matching the previous toss. So it is very much like tossing a fair coin and winning if you get (say) heads. The only difference is the "wasted" toss at the beginning. If we are tossing a fair coin, and $Y$ is the number of tosses until the first head, then $P(Y=n)=\frac{1}{2^n}$ ($n=1, 2, \dots$). But $X$ has the same distribution as $Y+1$. So $P(X=n+1)=P(Y=n)=\frac{1}{2^n}$, or equivalently $$P(X=n)=P(Y=n-1)=\frac{1}{2^{n-1}}\quad (n=2,3,\dots).$$
Generalization: Suppose that the probability of a head is $p$, and the probability of a tail is $q=1-p$. Either analysis of the problem can be extended to the more general situation. We win on the $n$-th toss if we did not win on the first $n-1$, and then the $n$-th toss matches the previous one.
It is convenient to split the analysis into two cases, $n$ odd and $n$ even. We do a full analysis of the odd case. So let $n=2m$. As is the first analysis, there are two ways to win on the $(2m+1)$-th toss. Either (i) we went HTHTHT and so on, with a tail on the $(2m)$-th toss, and then a tail again on the $(2m+1)$-th toss, or (ii) we went THTHTH and so on, with a head on the $(2m)$-th toss, and a head again on the $(2m+1)$-th toss.
The probability of (i) is $p^mq^m q$ and the probability of (ii) is $q^mp^mp$. Add them together. We get $p^mq^m(p+q)$, which is $p^mq^m$ ($m=1,2,\dots)$.
Next we deal with the case $n$ is even, say $n=2m$. The calculation is very similar to the previous one, so it will be omitted. The probability is the slightly more complicated-looking $p^mq^{m-1}p+q^mp^{m-1}q=p^{m-1}q^{m-1}(p^2+q^2)$, where $m$ ranges over the positive integers.
The (X,Z) outcomes are 00, 01, 02, 10, 11, 12, 13, 21, 22, 23. For each of these ten outcomes, determine how many results of the coin tosses generate it. Many correspond to only one result, for example 00 is generated by 0000 only hence its probability is 1/16. Others correspond to several results, for example 11 is generated by 1010, 1001 and 0100 hence its probability is 3/16. These ten probabilities should sum to 1. (To help you check your results, let me mention that six cases have probability 1/16, two have probability 2/16 and two have probability 3/16.)
Best Answer
(a)
If $B$ in your question denotes "binomial" then your answer is correct.
(b)
For convenience let $Z$ denote the number of tails obtained in the second toss.
Then $X-Z$ is the number of tails in the first toss, $Z$ is the number of tails obtained in the second toss and $Y-Z$ is the number of tails obtained in the last $2$ tosses.
This indicates that they are independent and also makes clear how they are distributed.
Now for every pair $\left(i,j\right)$ with $i\in\left\{ 0,1,2\right\} $ and $j\in\left\{ 0,1,2,3\right\} $ find $P\left(X=i,Y=j\right)$ on base of:
$$\begin{aligned}P\left(X=i,Y=j\right) & =\sum_{k=0}^{1}P\left(X=i,Z=k,Y=j\right)\\ & =\sum_{k=0}^{1}P\left(X-Z=i-k,Z=k,Y-Z=j-k\right)\\ & =\sum_{k=0}^{1}P\left(X-Z=i-k\right)P\left(Z=k\right)P\left(Y-Z=j-k\right)\\ & =\frac{1}{2}\sum_{k=0}^{1}P\left(X-Z=i-k\right)P\left(Y-Z=j-k\right) \end{aligned} $$ The marginals can be calculated by means of:
Actually the marginals are calculated in (a) already, so check them.