I play N times toss-the-coin game (try to guess the result of tossing a coin). The game is fair/unbiased, I have 0.5 probability to win in each game.
In N games I cannot sum probabilities (summation of probabilities rule) because 1/2 + 1/2 = 1 which is a stupid result (100% probability to win at least once in 2 games).
What is the correct way to calculate probability?
You cannot sum probilities because $P(A \cup B)=P(A)+P(B)-P(A\cap B)$ . That is, the probability of "$A$ or $B$" is the sum of the individual probabilities only if the events are disjoint (not the case here).
The event "winning at least one game" is the complement of "losing all games". Hence, calling $L_i$ the event "losing the game $i$", the desired probability is
$$1 - P(L_1 \cap L_2 \cap ... L_n)=1- \prod P(L_i) = 1- \prod(1-p_i)$$
where $p_i$ is the probability of winning game $i$. In this case we get $$1- \left(\frac12\right)^n$$