Lesson 14.1 Probability and the Basic Counting Principle

Vocab Breakdown: Probability: The likelihood that an event will occur (what happens, or what is thought to happen out of the total trials) Event: A set of desired results Tree Diagram: A pictorial depiction used to organize outcomes Counting Principle: If 1 event can happen m ways, and a 2nd event n ways, then both events can happen m x n ways Simple Event: An event with one outcome Independent Event: When occurrence of one event has no influence on other events Dependent Event: The probability of one event depends on the occurrence of another event

When Mr. Castillo got ready for work this morning, he found that he was in quite a predicament. He could not decide what to wear and even worse, he did not know how many choices he had. Here are his options: he has a maroon Perry polo, a white Perry polo, and a blue Perry polo. He has the choice of tan pants or black pants, and black shoes or brown shoes. How many possible outfits can Mr. Castillo make? (A picture or tree diagram might be helpful here!)

12 Possible Outfits! Answer:

Basic Tree Diagrams A tree diagram is simply a model of all the different outcomes that are possible. We saw an example of this in the warm-up when we drew all the possible clothing options Mr. Castillo had.

More on Tree Diagrams Making a tree diagram is a good option for small outcomes such as finding out how many types of sundaes you can make given 2 types of ice cream (chocolate and vanilla), 3 types of cones (waffle, sugar, and dipped), and 2 toppings (sprinkles and chocolate chips). 12 Ways!

But what happens if you need to figure out how many possible combinations can be formed if you have 4 sizes of pizza, 20 different sauces, and 40 different toppings?? A tree diagram would take a long time! This is where the Basic Counting Principle comes in…

Basic Counting Principle You and your friends are ordering a pizza. There are 4 types of meat, 2 types of cheese, and 5 types of veggies to choose from. How many different pizzas could you order? # of meats × # of cheeses × # of veggies 4 × 2 × 5 40 pizzas

Rule: The total number of outcomes is found by multiplying the number of choices for each stage of the event. Fundamental Counting Principle: If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m x n ways. (This continues for 3 or more events)

More Counting Principle! Ex. 1. How many different outfits could Jazmine pick out if she has 5 different jeans, 10 shirts, 6 pairs of socks/tights, and 12 pairs of shoes/boots? Ex. 2. To access a super secret spy facility you have to enter one of 5 elevators, then choose from 20 floors, after that you must stand under one of 4 colored pneumatic tubes. Mrs. Chesley is trying to break in, how many different possibilities are there for her to choose from in planning her evil scheme? 5 × 10 × 6 × 12 = 3600 Outfits 5 × 20 × 4 = 400

A Different Type of Problem Often times the counting principle is applied when trying to determine how many ways something can be done. Let’s look at some examples of this: If you have 7 toy cars that you want to put on display on a shelf at home, how many different ways can you arrange the cars? ___ ___ ___ ___ ___ ___ ___ = Think about how many spots there are Then think about how many choices of car you have for each spot.. Then multiply your answers to find the total… ways

More Practice If you are taking a picture of 6 of your friends standing in a single row, how many different ways can you arrange them? 6*5*4*3*2*1 = 720 ways You just recently got a job at the MVD and they want you to calculate how many possible combinations there are for the last 4 numbers on a license plate. Hint: Think about how many numbers you have to choose from for each spot! 10*10*10*10 = 10,000 combinations

Groupwork! In groups of 3 or 4 come up with your own problem that requires the use of the basic counting principle. Remember to be creative, and think about the types of problems that use the counting principle. When all groups are finished you will exchange the problems and let other groups solve yours, while you solve theirs. (Another option is to tape each group’s problem to the front board and have a race to see which group can correctly answer all of the problems!)

Homework: Complete the 14.1 Worksheet!