Question

solve number 6 only

Problem 6: (a) Using induction, derive inclusion exclusion formula for 3 events: PCAO BUC) = P(A)+F(B)+P(C) - PAB) - PANC) - P(BNC) + P(ANBAC). (b) Then, for n events: P(U4) - EPA) FAMA) "FAN (c) Using the above solve the following matching problem the deck of numbered cards is allocated only int e nden t and in onemlope). Find the probability that it lost here will match itsetvelopem ber Problem 7: total of 36 memborsota cu play tennis: 28 play squash, and I plan hadirinton Furthermore, the bottoms and all others and hp og de

Answer 1

problem 6 a). Given p CAU BUC) = p (4)+PⓇ+P10 - P(AOB) - P (AOC) - P(BOC+ Planbro proobo P (AUBUC) = PPAU (Buc)] = P W + P (BUC) - P (An tévely = p +[P®) + PlO – PCBNG)] - PE (ANB) u (Anc)] = P(A) + P B + PU- PCBNC) - PLANB) U (ANC) = P(A) + P(B+ P(C) – P (BA) - {P(ANB) & P (ANC) - P (AnBre)] PAI+PCB +PC)- P(ANB) - P(BOC) PENA) tp (AnBaC) -

- For n events. PCGA:)- pla:) - EP 19;naz) + PO NA tikken hp (A, NA₂ 1. An) - 0 proob: por two events A, and Az we have P CA, UA₂) = p (6) + 8 (42) - P (A, NA₂) Hence ① is true for n=2 2tah tet us now suppose that is true bar n=3. A ) = P(A) + P (A2) + P(As) - PA, naz) - P(A,na) - P(An A3 ) + P (A, .Az nAs) @ Now let's start with non so that PCtA;) = & P (A) I I p (A; IA;) tenet (+) hp (A, NA ...An) . * P * * * *:) = PS (1999) U Ami? = P ( 9 .) + P (*- P3 (8) P i 4 sjah Now , p(A) + P(A) - PS (An Anth} pla:) - EIP CA; ) Az ) + -- 4 kiden (blo (Anao - A) + P (Art) - PS 3 (A: A)} from a

h -) htl IP(A; ) - 2 P (A; DA; )+ --ft) p (An A₂ - A) isi Teisich -{& PCA, O Mus) in P (1939 1;9 ) +---+ |-yhlp (A1A0 -- An Ann San from ③ ) 5 P (8 A:) - 'p(0.-.. (41043) 4,3 P(4,0 4) tim-ot (oh p{ A, 9.6. -0,2} bu = p (A :) - IZ P(A;MA;) + - - *(-1) p (A, .. And) i lsichs (ht) Hence ib o is true bar non, it is also true bor n=2 Hence by the principle ob mathematical induction, it follows б і яе фоя оt роѕіvе іn еуол values of no
you can first problem following the solution of b. Try to solve c it's little blurry I was not getting it.