Probability for Coin toss

I know there are many coin toss problems out there. This however doesn't aim to find the probability, but the representation by the use of other pre-defined events. So here goes.

A coin is tossed N-times. Define a possible probability space for this experiment. We observe the events $A_{k}$ = "Tails at k-th toss" (k = 1,...,N), B = "Heads is never thrown" and C = "at least twice tails and at least once heads are thrown". Express the events B and C with the event $A_{k}$.

Have so far got $B = \bigcap_{k=1}^{N}A_{k}$ and $C = B^c \cap$ (exactly once tails U never tails). And for "never tails" i've got $(\bigcup_{k=1}^{N}A_{k})^c$. Any idea how to find "exactly once tails"?

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1 Answer

We have "exactly once tails" $\iff$ there is a $k$ such that tails happened precisely the $k$-th time (and only the $k$-th time) $\iff \exists k : (A_k \textrm{ and } (\forall j\neq k : A_k^c))$

Translating this to unions and intersections gives $$\bigcup_k \left( A_k \cap \left( \bigcap_{j\neq k} A_j^c \right) \right).$$

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In general this is a good way of translating events: first write down the event using quantifiers $\forall$ and $\exists$, then translate these to intersections and unions respectively. Example for event C: $\exists i : \exists j\neq i : \exists k : (A_i \textrm{ and } A_j \textrm{ and } A_k^c)$ translates to $\bigcup_i \bigcup_{i\neq j} \bigcup_k \left( A_i\cap A_j\cap A_k^c \right)$ – Daan Michiels Apr 24 '12 at 9:45