a A football team enters a tournamet in which they play one

(a) A football team enters a tournamet in which they play one game with each of three teams, but they get to choose the order in which they play their opponent teams. the team knows the probability of \"win\" against each of the three teams:

PA: is the probability to win when playing with team A

PB: is the probability to win when playing with team B

PC: is the probability to win when playing with team C

where pC<pB<pA

The team wins the tournament if they win 2 games in a row. The team wants to maximise the probability of winning. Show that the \"win\" probability is maximised if they play the weakest team second, and that the order of playing the other two teams doesnt matter.

(b) The team will only play with two teams: team A and team B, where pB<pA. Suppose that the team has a limited budget and they want to minimize the expected number of games they need to play to win two games in a row, should they start with A or with B? (justify)

HINT: let E[N₁] denote the mean number of games needed if they initially play i. Derive an expression for E[N_A] that involves E[N_B] ; write down the equivalent expression for E[N_B] and then subtract.

Solution

b. As it is given that pC<pB<pA so the winning probability will be high in case of A. playing with team A will be useful for the team as A is the weakest side than B & C. win over team A will give motivation and moral boost to team that they can also defeat team B.