In a fairground game there are two bags, each of which contains 4 coloured balls. There are a total of 4 red balls, 3 yellow balls and 1 blue ball. The player chooses one of the bags and removes two balls without replacing them. If the two balls are the same colour then the player wins. The player is equally likely to choose either bag and the balls are arranged to give the smallest possible probability for the player to win. What is the probability that the player wins?

Answer:

20) B

Since the balls are arranged in a way to give the smallest possibility for a player to win, the first bag must have 2 red balls and 1 yellow ball and 1 blue ball. The second bag must have 2 red balls and 2 yellow balls.

Probability of choosing either bag = 1/2

Let us say a player picked bag 1.
The only way to win is to pick 2 red balls.
Probability of picking a red ball = 2/4 = 1/2
In order to win, the player must pick a second red ball.
Probability of picking a second red ball = 1/3
Probability of picking two same coloured balls from Bag 1 = 1/2 x 1/2 x 1/3 = 1/12

Let us say a player picked bag 2.
Only way to win is to either pick 2 red balls or 2 yellow balls.

1st ball
Probability of picking a first red ball = 2/4 = 1/2
Probability of picking a first yellow ball = 2/4 = 1/2

2nd ball
If the player picked a red ball first, they must pick a red ball again if they want to win.
Probability of picking a second red ball = 1/3

If the player picked a yellow ball first, they must pick a yellow ball again if they want to win.
Probability of picking a second yellow ball = 1/3

Probability of picking two same coloured balls from Bag 2 = 2( 1/2 x 1/2 x 1/3 ) = 1/12 x 2 = 2/12

There are three possibilities to win:
2 red balls from bag 1 = Probability of 1/12
2 red balls from bag 2 = Probability of 1/12
2 yellow balls from bag 2 = Probability of 1/12

Probability of winning = (1/12 x 3) = 3/12 = 1/4