Suppose, you have been flipping a fair coin and got coin head 5 times in a row. Now, what is the probability of getting 6th?

On the one hand, it is said that probability is 1/2. On the other hand, the probability that next toss is a part of 6 (or longer) head streak is 2^(-6). So, which answer is right?

I suppose this is philosophical question, as it's about foundations of mathematics.

  • 1
    Each coin toss has a 50/50 chance of landing heads/tails. See here: stats.stackexchange.com/a/136879 – Mr. Kennedy Jul 7 '18 at 23:35
  • 3
    Strikes me as entirely a mathematics question. I mean, if this is also philosophy, basically every basic fact of math is a "philosophical question". Vote to close. – Chelonian Jul 8 '18 at 2:26
  • Coin tossing is a "memoryless" Markov process, e.g., math.stackexchange.com/questions/116464/… (so the answer's 1/2) – John Forkosh Jul 8 '18 at 6:32
  • This may be an interesting question about the realism of probability, but if that you were trying to do here, you need to rephrase the question to reflect it better. – Yechiam Weiss Jul 8 '18 at 6:58

If you define that your coin is a "fair coin", there is always a 50/50 chance of its landing on head or tail on each toss, no matter when. There is 1/2^5 (=0.03125) chance that you get heads 5 times in a row, and that's what you did get in the tossing you mentioned.

The problem is how you know it's a fair coin. Theoetetically, there is no way to know it is. You must toss it infinite times to be 100% certain that it is fair or not. Practically, you can only get close to this ideal knowledge,, by tossing it as many times as you can, say 1,000 times, and it came out 50/50. (Yet, you can never be 100% sure that it is fair if the the number of tossings is finite.).

For an unknown coin hat comes out heads five time in a row but without a defined (or assigned, to be more correct) tossing probability, it may be fair coin, a 70% head coin, a 90% tail coin, etc. So, the probability of tossing is undefined for this unknown coin. Until you define ( assign, to be more correct) the coin's probability of tossing, the probability of tossing of that coin remains forever undefined.


Your question relates to the gambler's fallacy, which is a well-known confusion in probability and statistics. If you have an exchangeable sequence of tosses of a fair coin then the probability of a head is one-half each time, regardless of the previous tosses. This result occurs by definition, because you have assumed fairness of the coin and independence of the tosses. The more interesting question here is what you should infer if you don't assume fairness of the coin a priori, and instead allow for the possibility of bias.

There is a series of papers in the academic journal The Mathematical Scientist looking at prediction of Bernoulli outcomes like this in cases where the mechanism is designed to be fair, but there is a possibility of bias (see O'Neill and Puza 2005, O'Neill 2012, and O'Neill 2015). The basic idea is that if you believe the observable outcomes to be exchangeable, and you don't know the direction of bias a priori, then you should predict whatever came up the most. Reading those papers will give you a set of probability results for this, as well as discussing the gambler's fallacy.


The correct anawer is 1/2. The 2^-6 probability applies to the total tosses, not to the individual flip. Furthermore getting the exact string heads-heads-heads-heads-heads-tails also has a probability 2^-6 which again shows that each flip has a 1/2 chance.

Not the answer you're looking for? Browse other questions tagged or ask your own question.