Assuming everything is fair what are the odds that one of the two sides in a coin toss wins 6 times in a row within the first 6 tosses?
Please also answer for the general case n times in a row within the first n tosses and prove or disprove that it is equivalent to:
Given a perfectly random coin toss what are the odds that the coin lands on the same side n times in a row within the first n tosses?
Since there are 2 possible outcomes in a coin toss, heads and tails, the probability of getting 6 heads in a row is $\Large\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{2^6}$. Similarly, probability of 6 tails would also be $\Large\frac{1}{2^6}$. And since we are looking for a row of 6 heads OR tails, we add up the probabilities to get the required answer, which is $\Large \frac{1}{2^5}=\frac{1}{32}$
Now you can easily deduce the generalisation to n times in a row which is $\Large\frac{1}{2^{n-1}}$