Question

Use a tree diagram to find the simple events for the experiments in Exercises \(21-24 .\) A bowl contains five candies-red, brown, yellow, blue, and orange. Draw two candies at random, one for you to eat, and one for a friend.

Short answer

Answer: There are 20 simple events in this experiment.

Step-by-step solution

Create a tree diagram

Begin by drawing a tree diagram, which consists of nodes and branches. The first node should represent the starting point (the given bowl of candies). From this node, draw five branches representing each possible candy color we could choose for ourselves (red, brown, yellow, blue, and orange).

Draw the branches for the friend's candy choices

For each choice we made in step 1, our friend can choose from the remaining four candies. Draw four branches for each of the five initial branches, representing each possible candy our friend could select. After we chose a candy, there will be only four candies left, so this step should result in 20 branches.

Label the branches with the chosen candy colors

On each of the branches, write the candy color that corresponds to the choice we or our friend made. For example, if the first branch in step 1 represents us choosing a red candy, and the four branches in step 2 represent our friend choosing a brown, yellow, blue, or orange candy, then label them accordingly.

Identify the simple events

Simple events refer to the unique outcomes in this experiment. In our tree diagram, each of the branching paths that end with our friend choosing a candy represents a simple event. Since there are 20 end branches in the tree diagram, there will be 20 simple events. These simple events are: 1. Red - Brown 2. Red - Yellow 3. Red - Blue 4. Red - Orange 5. Brown - Red 6. Brown - Yellow 7. Brown - Blue 8. Brown - Orange 9. Yellow - Red 10. Yellow - Brown 11. Yellow - Blue 12. Yellow - Orange 13. Blue - Red 14. Blue - Brown 15. Blue - Yellow 16. Blue - Orange 17. Orange - Red 18. Orange - Brown 19. Orange - Yellow 20. Orange - Blue This tree diagram represents all the simple events for the given experiment of drawing two candies at random, one for us and one for our friend.

Key concepts

Simple Events

In the context of probability, a simple event is an outcome that cannot be broken down into smaller parts. It is a single occurrence within an experiment that doesn’t share its outcome with any other event. When considering a tree diagram in probability, such as in the candy bowl example, every path from the starting point to an endpoint represents a separate simple event. This endpoint corresponds to a unique pair of candy selections that cannot occur simultaneously with any other selection.

For instance, the event 'Red - Yellow' signifies one has picked a red candy and the friend a yellow one, and this specific outcome is independent of all the other combinations of colors picked. The strength of using tree diagrams lies in its visual presentation of all possible simple events, which is especially useful when dealing with a limited number of outcomes, like in combinatorial experiments.

Probability Experiments

Understanding Probability Through Experiments

Probability experiments are structured ways to produce and observe outcomes, allowing us to study randomness and chance. In our candy example, the experiment involves drawing two candies from a bowl in succession, without replacement. The act of drawing candies constitutes a random experiment, as each outcome is uncertain and subject to variability.

Applying Tree Diagrams

Tree diagrams serve as a powerful tool in visualizing the various outcomes of probability experiments. They map out all possible outcomes, which are the simple events, systematically. With the branches clearly indicating the sequence of choices, the tree diagram helps to elucidate the structure of the experiment, making probability calculations more intuitive.

Combinatorics

The field of combinatorics delves into counting, arranging, and structuring elements within a set in an orderly way. It’s a critical aspect of probability that deals with the different ways objects can be selected or arranged, following specific rules. In our candy choosing exercise, combinatorics allows us to determine the number of possible ways two candies can be drawn from the bowl.

The fundamental principle of combinatorics called the counting principle can be applied here. It states that if one event can occur in 'm' ways and a subsequent event can occur in 'n' ways, then the total number of ways both events can occur is 'm' times 'n'. In this case, since there are five options for the first candy and four for the second (after the first is taken), there are in total 5 x 4 = 20 different possible pairs of candies that can be drawn, symbolizing the 20 simple events identified in the tree diagram.