Question

How does the theoretical probability of the event "flip heads" change when a coin is flipped more times in an experiment?

Answer 1

It doesn't change. The theoretical probability of "heads" is a property of the coin (and the flipping process) — for a fair coin P(heads) = 1/2, for a biased coin P(heads) = p — and that probability stays the same no matter how many times you flip.

What does change as you flip more is how the outcomes behave:

- Expected number of heads after n flips: E[X] = n·p. So the expected count of heads grows linearly with n.
- Distribution of the count: X ~ Binomial(n, p), Var(X) = n p (1−p).
- Proportion of heads (X/n) has mean p and standard deviation sqrt(p(1−p)/n), so the variability of the observed fraction decreases as n increases (roughly like 1/√n).

Thus, by the law of large numbers, the observed fraction of heads will tend to get closer to the theoretical probability p as the number of flips grows, even though the theoretical probability itself remains constant. Also note individual flips are (usually) independent, so past outcomes don’t change future probabilities (no “memory”).