11.4 Tree Diagrams 691 At the coffee station at work, you have several options when making a cup of coffee. You can choose to add no cream, regular cream, or hazelnut-flavored cream. You can also choose to add no sweetener, sugar, or aspartame. How many different ways can you prepare yourself a cup of coffee? If you were to randomly prepare a cup of coffee, what is the probability that you prepared a cup of coffee with regular cream and sugar? In this section, we will answer these questions using an important formula called the fundamental counting principle. We will also use a helpful diagram called a tree diagram to help us analyze the possible outcomes when multiple experiments are involved in problems. SECTION 11.4 Tree Diagrams LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand the fundamental counting principle. 7 Construct tree diagrams to determine probabilities. Why This Is Important Being able to determine the number of outcomes when multiple experiments are being performed is essential to our understanding of probability concepts. Throughout this section we will explore many real-world applications that involve both the fundamental counting principle and tree diagrams. Such applications may appear in a wide variety of occupations. The Fundamental Counting Principle We now introduce the fundamental counting principle. We will also use the fundamental counting principle in Section 11.7. Fundamental Counting Principle If a first experiment has M distinct outcomes and a second experiment has N distinct outcomes, then the two experiments in that specific order have ⋅ M N distinct outcomes. If we wanted to determine the number of possible outcomes when a coin is tossed and a die is rolled, we could reason that the coin has two possible outcomes, heads and tails. The die has six possible outcomes: 1, 2, 3, 4, 5, and 6. Thus, the two experiments together have ⋅ 2 6, or 12, possible outcomes. Example 1 Drawing Two Cards If two cards are drawn from a standard 52-card deck, determine the number of possible outcomes if the cards are drawn a) with replacement. b) without replacement. Solution a) With replacement means that after the first card is drawn, it is replaced in the deck. The first card drawn can be any one of the 52 cards in the deck. Since the first card is then replaced in the deck, the second card can also be any one of the 52 cards in the deck. Using the fundamental counting principle, we have the following. = ⋅ = with Number of ways to draw two cards replacement 52 52 2704 Antonio Guillem/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=