Event A means that you will be late to the lecture. Event B means that the lecturer will be late. What events must be added for this system of events to become a $\sigma$ algebra? My thoughts: I think that events "both will be late" and "neither of them will be late" must be added, but probably it's not enough. Would an event "lecturer won't be late" make this a $\sigma$ algebra?
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See also here – binkyhorse Jun 10 '14 at 18:27
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If $A$ and $B$ belong to a (sigma-)algebra, then so do $A\cap B$, $A^c\cap B$, $A\cap B^c$ and $A^c\cap B^c$. These sets are disjoint and there union is the whole space. If they are not empty then they form a partition of the space and every set that can be written as a union of these sets also belongs to it. That would give you $2^4=16$ sets in total.
drhab
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