Exercise 7. (20 points) The aim of this exercise is to prove that PE EU F) 2 PEF (a) Show that PEEUF) - PEFEUF))PFEUF) - PLEFOEUF) PEUT Hint: Use with EAEUF and E) - Flo forumis P(E(A) PEEA)PLEA) + PELLE) PLESA) (6) Show that und PEFEUF)) - PEF P(EFEUF)) = PLEFE) = 1 (c) Deduce that from (a) and (b) that PEEUF) - P(EF)PFEUF) + PLEEUF) (a) Use the formula P(FEUF) - 1-PIPE F) in (e), to show that P[EEUF) -- PLE\F)+(1 PLE\F))P[FEUF) (c) Deduce from the derived formuln in (d) that PEEUF) 2 PLE/F). (0) Show that P(E|BUF) - P(EF(EUF)PF EUF) + PCE FEUF)PFEUF) Hint: Use with E1 - EA-EUF am E - F this forums P(5:14) - PEJE,APIEA) + PERE A)PESA) (b) Show that P(EFEUF)) = PEF) and PELFEU F)) = P(EFE) = 1 (e) Deduce that from (a) and (b) that P(E\EU F) = P(EF)PFEU P) + P(FEUF) (d) Use the formula P(F|EU F) = 1 - P(FEU F) in (e), to show that P(EEUF) = P(EF) + (1 - P(EF))P(FOEUF) (e) Deduce from the derived fortuln in (d) that P(E EU F) 2 PLE/F)