A green yellow and red light are linked to a button When the

A green, yellow, and red light are linked to a button. When the button is pushed either the green, yellow, or red light will light up and will do so independently of all other outcomes. A game is created using this system.

You win the game by either pushing a button once and getting a green light, or pushing the button twice and getting either 2 green lights or exactly one red light. The probability of having the green light on a single push is constant and three times the probability of a red light. Find the probability of winning is the same for either the one-push or two-push option.

Solution

P( green light on a single push ) = 3 P( red light)

P( green ) = 1/3

P( green 2 times) = 1/3*1/3=1/9

1/3 + 1/9 = 4/9