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 • n c number of carriers • r number of roles • x number of carriers chosen by the Mastermind • c i ={ 1 if player i is carrier 0 otherwise • t a j i u combination of a j i over i • R k i i = "The player i has role k i " • M = "A Mastermind is in the game" • s i probability role i is survivor • p i “ 1 ́ s i • x ̈y r ́ 1 average over the roles excluding the Mastermind 1 3 Calculations E r n c s “ n ÿ i “ 1 ÿ t c j u c i P ́ t c j u ̄ “ “ n ÿ i “ 1 ÿ t c j u ÿ t R kl l u c i P ́ t c j u ˇ ˇ ˇ t R k l l u M ̄ P ́ t R k l l u M ̄ ` ` n ÿ i “ 1 ÿ t c j u ÿ t R kl l u c i P ́ t c j u ˇ ˇ ˇ t R k l l u M ̄ P ́ t R k l l u M ̄ A ” n ÿ i “ 1 ÿ t c j u ÿ t R kl l u c i P ́ t c j u ˇ ˇ ˇ t R k l l u M ̄ P ́ t R k l l u M ̄ “ “ n ÿ i “ 1 ÿ c i ÿ t R kl l u c i P ́ c i ˇ ˇ ˇ t R k l l u M ̄ P ́ t R k l l u M ̄ “ “ n ÿ i “ 1 ÿ t R kl l u P ́ c i “ 1 ˇ ˇ ˇ R k i i ̄ P ́ t R k l l u M ̄ “ “ n ÿ i “ 1 r ́ 1 ÿ k i “ 1 ÿ t R kl l ‰ i u p k i P ́ t R k l l ‰ i u ˇ ˇ ˇ R k i i M ̄ P ́ R k i i ˇ ˇ ˇ M ̄ P ́ M ̄ “ “ n ÿ i “ 1 r ́ 1 ÿ k i “ 1 p k i P ́ R k i i ˇ ˇ ˇ M ̄ P ́ M ̄ “ “ 1 r ́ 1 r ́ n r n ÿ i “ 1 r ́ 1 ÿ k i “ 1 p k i “ “ r ́ n r n x p y r ́ 1 2 B ” n ÿ i “ 1 ÿ t c j u ÿ t R kl l u c i P ́ t c j u ˇ ˇ ˇ t R k l l u M ̄ P ́ t R k l l u M ̄ “ “ n ÿ i “ 1 r ÿ k i “ 1 ÿ t R kl l ‰ i u P ́ c i “ 1 ˇ ˇ ˇ R k i i M ̄ P ́ t R k l l ‰ i u ˇ ˇ ˇ R k i i M ̄ P ́ R k i i ˇ ˇ ˇ M ̄ P ́ M ̄ “ “ n r n ÿ i “ 1 r ÿ k i “ 1 P ́ c i “ 1 ˇ ˇ ˇ R k i i M ̄ P ́ R k i i ˇ ˇ ˇ M ̄ “ “ n r ̃ x ` 1 ` n ÿ i “ x ` 2 r ́ 1 ÿ k i “ 1 p k i P ́ R k i i ˇ ˇ ˇ M ̄ ̧ “ “ n r ́ x ` 1 ` p n ́ x ́ 1 qx p y r ́ 1 ̄ E r n s s “ A ` B “ n ́ x p y r ́ 1 ` x ` 1 r p 1 ́ x p y r ́ 1 q ̄ “ “ n ́ 1 ́ x s y r ́ 1 ` x ` 1 r x s y r ́ 1 ̄ “ “ n ́ 1 ́ x s y ` x r ́ 1 x s y ̄ “ n ́ 1 ́ x s y ́ 1 ́ x r ́ 1 ̄ ̄ 3