# Is there a way to calculate the probabilities without tree diagram

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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I can´t understand,is it:"Distributive property of $\cup over \cap$"? –  HipsterMathematician Jul 22 '12 at 0:15
@MeAndMath, yes –  Hawk Jul 22 '12 at 0:23

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?