I have been doing some questions from an exam review with no solution and I have no idea how to work this problem. I know that $Pr(A_1) = \frac{1}{2}$, $Pr(A_2) = \frac{1}{2}$, $Pr(A_3) = \frac{1}{2}$, $Pr(A_1 \cap A_2) = \frac{1}{4}$, $Pr(A_1 \cap A_3) = \frac{1}{4}$, $Pr(A_2 \cap A_3) = \frac{1}{4}$ but how do I use this information?

Let's look at a tetrahedron (die with 4 faces). Each one of these faces has one of the bitstrings "110", "101", "011", "000".

For k=1,2,3, define the event

Ak = "the bitstring written on the bottom face has 0 (zero) at position k" For example, if the bitstring at the bottom face is 101, then A1 is false, A2 is true, and A3 is false.

Are the events A1, A2, A3 pairwise independent? Justify your answer.

Are the events A1, A2, A3 mutually independent? Justify your answer.

  • $\begingroup$ en.wikipedia.org/wiki/Pairwise_independence $\endgroup$ – Alex Mar 20 '14 at 17:04
  • $\begingroup$ In this link there's a link to the definition of mutual independence. This fully solves your question. $\endgroup$ – Alex Mar 20 '14 at 17:08
  • $\begingroup$ @Alex thats soo strange the example in your link is almost exactly the same question thanks $\endgroup$ – James Mar 20 '14 at 17:09
  • $\begingroup$ you are welcome $\endgroup$ – Alex Mar 20 '14 at 17:12
  • $\begingroup$ Essentially a duplicate of this question which is already on hold. $\endgroup$ – Dilip Sarwate Mar 20 '14 at 17:12

Two events are independent exactly if $$ \mathbb{P}(X \cap Y) = \mathbb{P}(X) \cdot \mathbb{P}(Y) \text{.} $$

Since per your question, $\mathbb{P}(A_1) = \mathbb{P}(A_2) = \frac{1}{2}$ and $\mathbb{P}(A_1 \cap A_2) = \frac{1}{4}$, the requirement above is satisfied (because indeed $\frac{1}{2}\frac{1}{2} = \frac{1}{4}$), and therefore $A_1$ and $A_2$ are independent.

For mutual independence, you need for events $E_1,\ldots,E_n$ that $$ \mathbb{P}(X_1 \cap X_2 \cap \ldots X_m) = \mathbb{P}(X_1) \cdot \mathbb{P}(X_2) \cdot \ldots \cdot \mathbb{P}(X_m) $$ for every selection of events $X_1,\ldots,X_m$ from $E_1,\ldots,E_n$.


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