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Alice and Bob have one coin each with probability of Heads $p$ and $q$, respectively. In each round, both Alice and Bob independently toss their coin once, and the game stops if one of them gets a Heads and the other gets a Tails. If they both get either Heads or both get Tails in any round, the game continues. Let $R$ be the expected number of rounds by which the game stops. Which of the following is $\text{TRUE?}$

  1. $R=\frac{1}{p+q-2 p q}$
  2. $R=\frac{1}{p+q-p^{2} q^{2}}$
  3. $R$ is independent of $p$ and $q$
  4. $R=\frac{1}{1+2 p q-p-q}$
  5. None of the above
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