Question

You enter a chess tournament where your probability of winning a game is 0.3 against half the players (call them type 1), 0.4 against a quarter of the players (call them type 2), and 0.5 against the remaining quarter of the players (call them type 3). You play a game against a randomly chosen opponent. What is the probability of winning

Answer 1

The probability of winning is 0.44 .

Explanation:

Here it's given that , You enter a chess tournament where your probability of winning a game is 0.3 against half the players (call them type 1), 0.4 against a quarter of the players (call them type 2), and 0.5 against the remaining quarter of the players (call them type 3). You play a game against a randomly chosen opponent. More precisely :

`Probability(type1) = 0.3\\Probability(type2) = 0.4\\Probability(type3) = 0.5\\\\Probability(winning) = w`

Now, we choose a random opponent and we need to find probability of winning which is possible as :

1. player chosen from type1

2. player chosen from type2

3. player chosen from type3

Combining all cases we get :

⇒
`w = type1(1-type2)(1-type3) + (1-type1)type2(1-type3) + (1-type1)(1-type2)type3`

⇒
`w = 0.3(0.6)(0.5)+0.7(0.4)(0.5)+0.7(0.5)(0.6)`

⇒
`w = 0.44`

∴ The probability of winning is 0.44 .