Construct a sample space $\Omega$ Construct a sample space $\Omega$ and events $A_1,\ldots, A_n$ ($n\ge2$) such that $\operatorname{Pr}(A_i) = \frac12$ ($1 \le i \le n$), every $n-1$ of the $A_i$ are independent, but the $n$ events are not independent.
Please help, I stuck for hours on this one.
 A: Consider $n-1$ independent symmetric random signs $(X_k)_{1\leqslant k\leqslant n-1}$. These are independent random variables such that $P[X_k=-1]=\frac12$ and $P[X_k=1]=\frac12$ for every $1\leqslant k\leqslant n-1$. Let $X_n=X_1X_2\cdots X_{n-1}$, so that $X_n$ is also a symmetric random sign. Then a solution to your problem is to define, for every $1\leqslant k\leqslant n$, $A_k=[X_k=1]$. This works for every $n\geqslant2$ (can you show it does?).
The events $(A_k)_{1\leqslant k\leqslant n}$ are not independent since $A_1\cap A_2\cap\cdots\cap A_n=A_1\cap A_2\cap\cdots\cap A_{n-1}$ hence $P[A_1\cap A_2\cap\cdots\cap A_n]=\left(\frac12\right)^{n-1}$ while $P[A_k]=\frac12$ for every $1\leqslant k\leqslant n$ hence $P[A_1]P[A_2]\cdots P[A_n]=\left(\frac12\right)^{n}$.
About the sample space: I strongly advise to leave $\Omega$ unspecified, as is done above. However, if your teacher insists (for which pedagogical reasons I would love to know), a solution is to consider the discrete hypercube $\Omega=\{0,1\}^{n-1}$ endowed with the sigma-algebra $2^\Omega$ and the uniform probability $P$, and to define $A_n=\{\omega\in\Omega\mid\omega_1+\omega_2+\cdots+\omega_{n-1}\ \text{even}\}$ and $A_k=\{\omega\in\Omega\mid\omega_k=1\}$ for every $1\leqslant k\leqslant n-1$.
