Carrier frequency in a Left 4 Dead game (WW) Captain Cap October 15, 2020 1 Assumptions • Probability of role repeats is negligible • Roles are included in the game with equal probability • The number of carriers chosen by the Mastermind is constant • Omitting the consequences of Family’s inclusion gives a negligible error (!) 2 Notation • n number of players • nc number of carriers • r number of roles • x number of carriers chosen by the Mastermind • ci ={ 10 if player i is carrier otherwise • taji u combination of aji over i • Riki = "The player i has role ki " • M = "A Mastermind is in the game" • si probability role i is survivor • pi “ 1 ´ si • x¨yr´1 average over the roles excluding the Mastermind 1 3 Calculations n ÿ ÿ ´ ¯ Ernc s “ ci P tcj u “ i“1 tcj u n ÿ ÿ ÿ ´ ˇ ¯ ´ ¯ ci P tcj uˇtRlkl uM P tRlkl uM ` ˇ “ i“1 tcj u tRkl u l n ÿ ÿ ÿ ´ ˇ ¯ ´ ¯ ci P tcj uˇtRlkl uM P tRlkl uM ˇ ` i“1 tcj u tRkl u l n ÿ ÿ ÿ ´ ˇ ¯ ´ ¯ A” ci P tcj uˇtRlkl uM P tRlkl uM “ ˇ i“1 tcj u tRkl u l n ÿ ÿ ÿ ´ ˇ ¯ ´ ¯ ci P ci ˇtRlkl uM P tRlkl uM “ ˇ “ i“1 ci tRkl u l n ÿ ÿ ´ ˇ ¯ ´ ¯ P ci “ 1ˇRiki P tRlkl uM “ ˇ “ i“1 tRkl u l n r´1 ÿ ÿ ÿ ´ ˇ ¯ ´ ˇ ¯ ´ ¯ kl ˇ ki pki P tRl‰i uˇRi M P Riki ˇM P M “ ˇ “ i“1 ki “1 tRkl u l‰i n r´1 ÿ ÿ ´ ˇ ¯ ´ ¯ pki P Riki ˇM P M “ ˇ “ i“1 ki “1 n r´1 1 r´n ÿ ÿ “ pk “ r ´ 1 r i“1 k “1 i i r´n “ nxpyr´1 r 2 n ÿ ÿ ÿ ´ ˇ ¯ ´ ¯ B” ci P tcj uˇtRlkl uM P tRlkl uM “ ˇ i“1 tcj u tRkl u l n ÿ ÿ r ÿ ´ ˇ ¯ ´ ˇ ¯ ´ ˇ ¯ ´ ¯ kl ˇ ki P ci “ 1ˇRiki M P tRl‰i uˇRi M P Riki ˇM P M “ ˇ ˇ “ i“1 ki “1 tRkl u l‰i n r nÿ ÿ ´ ˇ ¯ ´ ˇ ¯ P ci “ 1ˇRiki M P Riki ˇM “ ˇ ˇ “ r i“1 k “1 i ˜ ¸ n r´1 n ÿ ÿ ´ ˇ ¯ ki ˇ “ x`1` pki P Ri ˇM “ r i“x`2 k “1 i n´ ¯ “ x ` 1 ` pn ´ x ´ 1qxpyr´1 r ´ x`1 ¯ Erns s “ A ` B “ n xpyr´1 ` p1 ´ xpyr´1 q “ r ´ x`1 ¯ “ n 1 ´ xsyr´1 ` xsyr´1 “ r ´ x ¯ ´ ´ x ¯¯ “ n 1 ´ xsy ` xsy “ n 1 ´ xsy 1 ´ r´1 r´1 3
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