Roulette with p=$\frac{2}{3}$. What is the probability of not going home?

Question

I'm learning about the gamblers ruin. The problem is that I don't know how to calclate the formula.

I got two exercise questions in my book. Both of the questions will be about a strange roulette game. The probability of red in this game is $\frac{2}{3}$. In both of the games I will start with 1000. The player always bet on red, because of the high p. He will only spend 1 in each round. If he wins he will get 2 and if he lose he will lose 1.

First question:

In the first question the player is going home if he reaches 10^10 or if the
player has no money left. What is the probability that the player will never go home?

Second question

In this question the player will only go home, if the player has no money 
left. What is the proability of never going home?

Answers

First question: By just looking at the question I know the probability of not going home is 0, because you will either lose all your money or you will get 10^10.

I used the gambler ruin formula for this question:

$$ \frac{1-(\frac{1-p}{p})^{i}}{1-(\frac{1-p}{p})^{n}} $$ So I filled in: $$ \frac{1-\frac{2}{3}}{\frac{2}{3}} = \frac{1}{2} $$

$$ \frac{1-(\frac{1}{2})^{10^{3}}}{1-(\frac{1}{2})^{10^{10}}} + \frac{1-(\frac{1}{2})^{10^{10-3}}}{1-(\frac{1}{2})^{10^{10}}} = 0$$

Second answer is $$ 1 - \frac{1-(\frac{1}{2})^{10^{10-3}}}{1-(\frac{1}{2})^{10^{10}}} $$

At both of the question I do not know how to calculate them, because the exponents are too high. And because I can't calculate it, I have no idea if my answers are correct.

Are there any tricks to calculate? And are my answers correct?

Answer 1

If you're OK with an approximate answer, then you could use the binomial expansion and drop "small" terms appropriately: \begin{align} 1 - \frac{1 - 2^{-10^7}}{1 - 2^{-10^{10}}} &\approx 1 - (1 - 2^{-10^7})(1 + 2^{-10^{10}} ) \\ &= 1 - \left( 1 - 2^{-10^7} + 2^{-10^{10}} - 2^{-(10^7 + 10^{10})} \right) \\ &\approx 1 - \left( 1 - 2^{-10^7} \right) \\ &= 2^{-10^7}. \end{align} This would be perfectly acceptable in my field (physics), but the levels of rigor required in your might vary.