Redo Example 1 using Formula (1). 

Approach We will define the events as we did in Example 1. We want the probability of selecting a black ball. This can occur by selecting urn I and then selecting a black ball or by selecting urn II and then selecting a black ball. With B representing the event “black ball selected,” we have

Formula 1

Data from Example 1

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?

where P(B) = P(BU) + P(BM U) = P(U) P(BU) + P(U2) P(B|U) P(U) = 1/13 23 P(U2) = 4 P(B|U) = P(B|U) = 7 13