Suppose that there are two urns. Urn I contains four black and three white balls. Urn II contains four black, seven white, and two red balls. Suppose that we roll a fair die. If the die is a one or a two, we randomly select a ball from urn I. However, if the die is a three, four, five, or six, we randomly select a ball from urn II. What is the probability that the ball is black?

Approach We define the events as follows:

U₁: Urn I is chosen 

U₂: Urn II is chosen 

B: Black ball is chosen 

W: White ball is chosen 

R: Red ball is chosen 

We answer the question using a tree diagram.