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?
Best Answer
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.