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Could someone tell me how on earth did they calculate $P(A_1\text{ and }A_3)$ and etc.. without drawing a tree diagram?

Thank you very much

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    @MeAndMath, yes2012-07-22

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These are independent events and you want to know $\,P(A_1\cap A_3)=P(A_1)\cdot P(A_3)=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$ since I assume the probability of the coin being T or H is the same.

Btw, for any $\,1\leq i\neq j\leq 4\,\,,\,\,P(A_i\cap A_j)=P(A_1\cap A_3)=\frac{1}{4}\,$ . Why didn't you ask about these ones?