Jr jnag kybt_2(k) + (1 - k)ybt_2(1 - k) gb or nf fznyy nf cbffvoyr. Vg vagrefrpgf 0 ng k = 0 naq k = 1, fb jr pna svaq gur ybpny zvavzhz. Qvssreragvngvat, jr trg ybt_2(k) = ybt_2(1 - k). (Be lbh pna rkcbaragvngr gb trg evq bs gur ybtf). Fb k = 1/2. Fb lbh jnag gb pubbfr n frg gung pncgherf gur nf pybfr gb unys gur cebonovyvgl nf cbffvoyr.
Purnc nafjre: K pbafgnagyl gnxrf ba n fvatyr inyhr. Fb ragebcl vf mreb, ohg nyfb Trbetr xabjf jung vg jvyy or. Yrff purnc nafjre: n snve qvr. Ragebcl vf 2.5-vfu. Lbh unir gb nfx 3 dhrfgvbaf (yvxr bqq-if-rira naq gura ryvzvangr gjb bs gur 3 erznvavat va bar dhrfgvba rnpu).
Vs lbhe tebhcf ner ernyyl fxrjrq, gura lbh pna'g qb gur tebhcvat vagb unyirf irel jryy. Yvxr, vs lbh unir 99% bs lbhe cebonovyvgl ba bar nafjre, gura 100 zber rnpu jvgu 0.01% bs gur cebonovyvgl. Lbhe ragebcl vf yrff guna 1, ohg lbhe rkcrpgrq ahzore bs dhrfgvbaf vf terngre guna 1 (0.99 1 + 0.1 5).
Qrcraqvat ba ubj znal aba-1f lbh nyybj ng bapr va n irpgbe, lbh unir gb frg rnpu ryrzrag gb abg-1 n qvssrerag ahzore bs gvzrf. Vs lbh qba'g nyybj nal, gura bs pbhefr lbh trg 0. Fb jr pna nyjnlf unir ng yrnfg 1 irpgbe. Vs lbh nyybj bar, naq npg flzzrgevpnyyl, rnpu pbzcbarag vf abg-1 (a - 1) gvzrf. Vs gjb, gura (a - 1)(a - 2) gvzrf. Naq fb ba gb gur fhz bs (a - 1)!/(a - x)! sbe x orvat gur ynetrfg ahzore lbh unir ng bapr.
Va erghea, lbh trg gb rawbl n qvfgvapg irpgbe jvgu lbhe pbzcbarag ng 1 sbe gur fhz bs (a - 1)!/(a - 1 - x)! gvzrf. Fb jura guvf ahzore vf zber guna 3 gvzrf terngre guna gur ahzore orsber, lbh pna unir gur fhz sebz 0 gb x bs a-pubbfr-x irpgbef va gur frg.
V guvax. Gur nytbevguz znl jryy or gbb terrql (V arrq gb fubj lbh arire jnag gb fxvc frggvat n ahzore bs aba-mreb pbzcbaragf). Ohg V'ir glcrq guvf vagb n ebg13 obk naq lbh pna'g fnir n qensg pbzzrag fb V'yy fgbc gurer.
Guvf vf evtug, avpr! Gunaxf nyfb sbe gur pbeerpgvba.
Gur bayl pbzzrag V unir vf: sbe gur ynfg ceboyrz, lbh fubhyq or noyr gb jevgr qbja n pybfrq sbez, be ng yrnfg na rfgvzngr. Jung xvaq bs tebjgu engr qbrf guvf fhzzngvba unir? Cebonoyl Fgveyvat jvyy gnxr lbh pybfr gb gur evtug nafjre.
Urer'f nabgure jnl gb trg fbzrguvat: tebhc hc gur pbbeqvangrf va frgf bs guerr naq gnxr gur frg bs nyy irpgbef juvpu unir (1,1,1),(0,1,1),(1,0,1),(1,1,0) va rnpu gevcyr bs pbbeqvangrf. Guvf rknzcyr tvirf lbh 2^{(2/3)a} irpgbef.
Whfg gb yrg lbh xabj, V'z jbexvat ba guvf! V guvax vg ybbxf ebhtuyl rkcbaragvny orpnhfr lbh'er trggvat xvaq bs a!/(a/3)!. Abg gbb pbasvqrag gubhtu
As usual, post solutions in ROT13.