Isbarbaryaaca Simpson waa mid ka mid ah waxyaabaha si fudud loo fahmi karo isla markaana isla markaa layaabka ku leh dhacdooyinka tirakoobka. Waxay dhacdaa mar kasta oo kooxaha xogtu muujiyaan isbeddel gaar ah, laakiin isbeddelkaas ayaa la rogaa markii kooxaha la isku daro. Iyada oo la adeegsanayo tusaale fudud, iskhilaafka ayaa isla markiiba la fahmi karaa.
Waxaan tixgelineynaa labada qaybood ee kala jaban \(\#1\) iyo \(\#2\) iyo sidoo kale \(G = \#1 \cup \#2\) waxaan tijaabineynaa heerka guusha ee \(A\) iyo nidaamyadan \(B\):
| \(A\) | \(B\) | \(win\) | |
| \(\#1\) | \(\frac{1}{1}=100\%\) | \(\frac{3}{4}=75\%\) | \(A\) |
| \(\#2\) | \(\frac{2}{5}=40\%\) | \(\frac{1}{3}=33\%\) | \(A\) |
| \(\#1 \cup \#2\) | \(\frac{3}{6}=50\%\) | \(\frac{4}{7}=57\%\) | \(B\) |
Waxay soo baxday in \(A\) uu ka guuleysan yahay \(B\) \(\#1\) iyo sidoo kale \(\#2\) \(B\) , laakiin si la yaab leh \(G\) \(B\) ka guul badan \(A\) . Tusaalahani sidoo kale waa mid ka mid ah kuwa leh kuwa ugu yar \(G\) leh \(|G|=13\) . Ma jiro \(G\) leh \(|G|<13\) (caddayn xoog xoog leh).
Waxaan hadda subdivide ballamay \(G\) halkii \(2\) galay \(3\) disjoint subsets \(\#1, \, \#2, \, \#3\) la \(\#1 \cup \#2 \cup \#3 = G\) . Kadibna waxaan \(e_k \neq \emptyset\) kiiska xiisaha leh ee shey kasta \(e_k \neq \emptyset\) awoodda la dejiyay \(P(G)\) ee \(G\) soo socda ayaa khuseeya: $$\forall e_1, e_2 \in P(G): |e_1| \neq |e_2| \Rightarrow win(e_1) \neq win(e_2) \land |e_1| = |e_2| \Rightarrow win(e_1) = win(e_2)$$ $$\forall e_1, e_2 \in P(G): |e_1| \neq |e_2| \Rightarrow win(e_1) \neq win(e_2) \land |e_1| = |e_2| \Rightarrow win(e_1) = win(e_2)$$
Dhawr saacadood ka dib xoog caayaan on Core i7 caadiga ah, tusaalaha soo socda ayaa laga heli karaa:
| \(A\) | \(B\) | \(C\) | \(win\) | |
| \(\#1\) | \(\frac{6}{7}=85,71\%\) | \(\frac{12}{15}=80,00\%\) | \(\frac{22}{37}=59,46\%\) | \(A\) |
| \(\#2\) | \(\frac{95}{167}=56,89\%\) | \(\frac{48}{88}=54,55\%\) | \(\frac{38}{67}=56,72\%\) | \(A\) |
| \(\#3\) | \(\frac{48}{144}=33,33\%\) | \(\frac{16}{50}=32,00\%\) | \(\frac{2}{20}=10,00\%\) | \(A\) |
| \(\#1 \cup \#2\) | \(\frac{101}{174}=58,05\%\) | \(\frac{60}{103}=58,25\%\) | \(\frac{60}{104}=57,69\%\) | \(B\) |
| \(\#1 \cup \#3\) | \(\frac{54}{151}=35,76\%\) | \(\frac{28}{65}=43,08\%\) | \(\frac{24}{57}=42,11\%\) | \(B\) |
| \(\#2 \cup \#3\) | \(\frac{143}{311}=45,98\%\) | \(\frac{64}{138}=46,38\%\) | \(\frac{40}{87}=45,98\%\) | \(B\) |
| \(\#1 \cup \#2\cup \#3\) | \(\frac{149}{318}=46,86\%\) | \(\frac{76}{153}=49,67\%\) | \(\frac{62}{124}=50,00\%\) | \(C\) |
Halkaas (iyadoo loo maleynayo waqti xisaabeed aan macquul aheyn) tusaalooyin \(n\) qaybo hoose oo isku mid ah oo leh dhaqan isku mid ah ayaa laga heli karaa. Haddii kiisaska noocan oo kale ah ay dhacaan dhab ahaan, gunaanad kasta oo ku saleysan talo soo jeedinta guusha kooxda waa kuwo macquul ah oo aan macno lahayn.
Waqtigan xaadirka ah, waxaan kugula talineynaa akhriska xiisaha leh Sababaha: Moodellada, Sababaynta iyo tixgelinta by Judea Pearl .