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I'm reading high school probability, and this doubt arose while reading about successive coin tosses.

Suppose I'm tossing an unbiased coin, and recording the outcomes. Let's say that by pure chance, I get 'n' consecutive heads. I'm confused about the probability of the (n+1)th coin toss.

When I asked my teacher this, she said that the probability of any (unbiased) coin toss will always be 1/2, regardless of the previous outcomes, as each toss is an independent event. This does make sense to me, but there's a further question.

Since the statistics have to balance out, i.e. after a considerable number of tosses the head-tail ratio should reach close to 1/2, isn't the probability of getting a tail also increasing, with each successive head that we get?

If not, and the probability of each coin toss is indeed 1/2, then what's changing as we keep getting heads?

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First things first, statistics doesn't have to balance out. It's not a force of nature, but a rigorous study of patterns.

Second, there are independent events and dependent events. Tossing a coin is independent of any past events. It's similar to picking up a ball each from $n$ different boxes with each box containing a black ball and a white ball. The probability of picking a white ball from $5th$ box is independent from the results from other boxes. It's always $1\over2$.

Now consider only one big box with $2n$ balls ($n$ white and $n$ black). You take a ball from the box and discard it outside. Now, the probability of getting white in $5th$ try would be dependent on previous outcomes.

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That's the same "dilemma" as for the "delayed" numbers in lottery.

If once you have obtained $n$ heads, and after that, you are going to bet for a consecutive head then you have a probability of $1/2$, the same you would get for whichever previous pattern.

The problem is completely different if, before starting the tosses, you bet that in the following $n$ tosses you are going to get $m$ heads consecutive. Then here you are going to consider the "statistics" of all the possible results in n tosses

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