Question

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In the game Starcraft there are three races: Terran, Zerg, and Protoss. A new player to the game gets to pick one race with equal probability. Once a race is chosen, the players will stay with their choice for their entire gaming career.

Based on the league record,

• 60% of new Terran players can get out of Bronze League,
• 30% of new Zerg players can get out of Bronze League, and• 90% of new Protoss players can get out of Bronze League.

Suppose that a new player just gets promoted from Bronze League. What is the chance that this player plays Zerg?

Answer 1

Let A,B and C be events that a randomly selected player plays Terran, Zerg and Protoss respectively.

And Y be the event that player gets promoted from Bronze League

P(A) = P(B) = P(C) = 1/3

P(Y|A) = 0.6, P(Y|B) = 0.3, P(Y|C) = 0.9

The required probability that the player plays Zerg given that he gets promoted from Bronze League = P(B|Y)

Using Baye's theorem,

P(B|Y) = P(Y|B)*P(B)/P(Y)

where P(Y) = P(Y|A)*P(A) + P(Y|B)*P(B) + P(Y|C)*P(C)

= 0.6*1/3 + 0.3*1/3 + 0.9*1/3 = 0.6

Thus, P(B|Y) = 0.3/(3*0.6) = 1/6