math probability

Assume that Pr[E]=0.55,Pr[F]=0.55,Pr[G]=0.55,Pr[E∪F]=0.85,Pr[E∪G]=0.8, and Pr[F∪G]=0.75.

Find:
Pr[E' U F]; Pr[F' ∩ G]; and Pr[E ∩ G]

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  1. It looks like a Venn diagram problem to me - which is a pity because I can't draw a Venn diagram here. Never mind: I'll try to manage with just the numbers. Pr(EuF) = P(E) + P(F) - P(EnF) = 0.55 + 0.55 - Pr(EnF) = 0.85, so Pr(EnF) = 1.10 - 0.85 = 0.25. Likewise Pr(EuG) = Pr(E) + Pr(G) - Pr(EnG) = 0.55 + 0.55 - Pr(EnG) = 0.80, so Pr(EnG) = 1.10 - 0.80 = 0.30 (which is the third of the three answers you need). Likewise Pr(FuG) = Pr(F) + Pr(G) - Pr(FnG) = 0.55 + 0.55 - Pr(FnG) = 0.75, so Pr(FnG) = 1.10 - 0.75 = 0.35. Also Pr(F' n G) is everything outside F that's in G, which is Pr(G) - Pr(FnG) = 0.55 - 0.35) = 0.2. Finally, Pr(E' u F) = Pr(E') + Pr(F) - Pr(E' n F) = (1-0.55) + 0.55 - (0.55-0.20) = 1 - 0.35 = 0.65. I'm not sure I've got the above right, so I suggest you draw your own diagram and see if you get the same answers as me.

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