Probability coin toss question

Question

In a 50:50 gambling game where you double your stake each time you guess the correct outcome, how many times harder would it be to guess the correct outcome 15 times in a row than 3 times in a row? I’m guessing it’s not 5x harder, right?

Thanks!

Answer 1

For guessing the outcome $15$ times in a row, it is $(\frac{1}{2})^{15} = \frac{1}{2^{15}}$. For guessing the outcome $3$ times in a row, it is $(\frac{1}{2})^{3} = \frac{1}{8}$. Therefore, the amount of times harder it would be to guess an outcome $15$ times in a row as opposed to $3$ times in a row is ${2^{12}} = \fbox{4096}$ times harder.