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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?

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    $\begingroup$ When do we win 6 times? $\endgroup$
    – callculus42
    Commented Feb 22, 2020 at 15:58
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    $\begingroup$ Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. $\endgroup$
    – saulspatz
    Commented Feb 22, 2020 at 16:14
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    $\begingroup$ If I understand your Question, you need to specify how many times you will try. With enough attempts the probability of the same side 6 times in a row becomes as close to 100% as we wish. If you plan to toss the coin exactly six times, that should stated. $\endgroup$
    – hardmath
    Commented Feb 22, 2020 at 16:18

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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 $\Large\frac{1}{2^{n-1}}$

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  • $\begingroup$ As in almost all discussions of this problem, there’s an unstated assumption that there are only six coin tosses in all; the framer of OP’s problem probably made the same assumption. But if there were 100 tosses in all, wouldn’t the probability of getting six heads or six tails in a row out of those hundred be very different? $\endgroup$
    – Lubin
    Commented Feb 22, 2020 at 17:09
  • $\begingroup$ Yeah definitely. I posted my answer based on the assumption that the total number of tosses is 6. $\endgroup$
    – Aniket Gupta
    Commented Feb 22, 2020 at 17:17

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