Combinatorial probability

7.1 What event does the union of two subsets correspond to?

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7.2 Prove that the probability of any event is at most 1.

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7.3 What is the probability of the event $E$ that we throw an even number with the dice? What is the probability of the event $T=\{3,6\}$ that we toss a number that is divisible by 3?

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7.4 Prove that if $A$ and $B$ are exclusive, then $P(A)+P(B)=P(A\cup B)$.

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7.5 Prove that for any two events $A$ and $B$,

$$P(A\cap B)+P(A\cup B)=P(A)+P(B).$$

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7.6 Which pairs of the events $E$, $O$, $T$, $L$ are independent? Which pairs are exclusive?

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7.7 Show that $\varnothing$ is independent from every event. Is there any other event with this property?

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7.8 Consider an experiment with sample space $S$ repeated $n$ times ($n\ge 2$). Let $s\in S$. Let $A$ be the event that the first outcome is $s$, and let $B$ be the event that the last outcome is $s$. Prove that $A$ and $B$ are independent.

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