Tree Diagrams Section 5

Counting Principle If one experiment has M different outcomes and another experiment has N different outcomes, then the combination of both experiments, one after the other, has M·N different outcomes. Example: At a restaurant, you may have 3 different choices for a main entree and 5 different choices for a side. In all, you have 3·5 = 15 different possibilities for meals!

Tree Diagrams Sometimes, when applying the counting principle, it can be difficult to determine the number of possibilities that can arise, based on how complicated the experiments are. It can be helpful to make a visual diagram to help list all of the possible outcomes of an experiment. Tree diagrams are often a helpful way of produce complete lists of the outcomes of the combinations of multiple experiments.

Steak Outcomes Fish Chicken Fries Salad Coleslaw Soup Baked Potato Steak + Fries Salad Steak + Salad Steak Coleslaw Steak + Coleslaw Soup Steak + Soup Outcomes Baked Potato Steak + Baked Potato Fries Fish + Fries Salad Fish + Salad Fish Coleslaw Fish + Coleslaw Soup Fish + Soup Baked Potato Fish + Baked Potato Fries Chicken + Fries Salad Chicken + Salad Chicken Coleslaw Chicken + Coleslaw Soup Chicken + Soup Baked Potato Chicken + Baked Potato

With or Without Replacement When performing the same experiment multiple times, there are two ways of doing so: With replacement - outcomes from one experiment are still available in the next experiment Examples: die rolls, coin flips, spinners, drawing cards from a deck and shuffling the card back in Without replacement - outcomes from early experiments won’t be available in later experiments Examples: drawing balls from urns and not putting them back in, drawing cards from a deck and leaving them out