Q:

Probability of 20 heads without getting 2 tails in a row, versus 10 heads without getting 1 tail

Are the odds of flipping a coin and getting 20 heads without getting 2 tails in a row the same as flipping a coin and getting 10 heads without getting 1 tail?

A:

For the second experiment, the probability is simply $\left(\frac12\right)^{10}$.
For the first experiment, divide it into $20$ experiments where you flip a coin, then if it lands tails you flip it again, and the individual experiments succeed unless they yield tails twice. Each successful experiment yields heads once. Thus the overall experiment succeeds exactly if all $20$ experiments succeed, and the success probability for each of them is $\frac34$, so the overall success probability is
$$\left(\frac34\right)^{20}=\left(\frac9{16}\right)^{10}\gt\left(\frac12\right)^{10}\;.$$

