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Prove that if E and F are independent ev...

Prove that if E and F are independent events, then so are the events E and F.

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Since E and F are independent, we have
`P(EcapF)=P(E).P(F)`
From the venn diagram, it is clear that `EcapF " and " E' cap F` are mutually exclusive events and also `F=(EcapF)cup(E'capF)`

Therefore, `P(F)=P(EcapF)+P(E'capF)`
or `P(E'capF)=P(F)-P(EcapF)`
=P(F)-P(E).P(F)
=P(F)[1-P(E)]
=P(F).P(E')
Hence, E' and F are independent.
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