Playing a dice game, where the winner is determined when the first 6 appears

Question

I did a course on probability in undergrad, and I felt like brushing up on it by working out some problems a colleague gave me:

"Two siblings are playing a dice game, where the winner is determined when the first 6 appears"

1. What's the probability that the 1st sibling wins?
2. What is the mean number of rolls needed before a winner is determined? (1 turn consists of 2 dice rolls, assuming no one won in that turn)
3. Is is true that they could settle the game by flipping a fair coin instead of rolling the dice?
4. Suppose that k dice are now rolled simultaneously by each player on his turn, and now the first player to obtain a total of k (or more) 6’s, accumulated over all his throws, wins the game. (For example, if k = 3, then player 1 will throw 3 dice, and keep track of any 6’s that show up. If player 1 did not get all 6’s then player 2 will do the same. Assuming that player 1 gets another turn, he will again throw 3 dice, and any 6’s that show up will be added to his previous total.) Should you expect this game to run for fewer turns than the original game? Compute the expectation

My attempt

1. Let $X$ be a random variable indicating the probability of getting a six. Now, $X\sim Geo(m)$, where $m$ represents the number of failures. Then, $Pr[X=x]=(1-\frac{1}{6})^m\frac{1}{6}$.
2. Since the geometric distribution has an expectation of $\frac{1}{p}$, then the mean number of rolls will be 6
3. For a fair coin the probability of getting heads or tail would be $\frac{1}{2}$. By virtue of this the random variable, $Y$, representing this wouldn't be the same i.e $Pr[Y=y]=(1-\frac{1}{2})^m\frac{1}{2}$ and the expectations would also be different. Thus it is not true.
4. I'm a little clueless about this one. But I believe it should use the negative binomial distribution somewhere here

Answer 1

For question 3: if you flip a coin 3 times and write the outcome in binary you get a number 0-7 uniform random. Redo if you get a 0 or 7 else you have a die roll.

For question 1: do you want the probability he wins if he goes first? If yes then this is the probability first 6 is on an odd roll.