Probability & Statistics The Counting Principle Section 12-1

Independent Events  Independent events When the occurrence of one event does not affect the occurrence of another event. Example 1 event 1: a sunny day event 2: a Tuesday A sunny day will have no bearing on whether it is a Tuesday. The fact that a day is a Tuesday will not impact whether it will be sunny. Example 2 From a deck of 52 playing cards a person randomly selects 1 card, replaces it, and selects another card.

Important Definitions  Dependent events When the occurrence of one event affects the occurrence of another event. Example 1 event 1: A snowy winter event 2: people skiing If we have a snowy winter, that will affect the number of people who go skiing. Example 2 From a deck of 52 playing cards a person randomly selects 1 card, without replacing it, and selects another card.

Using the Counting Principle  You are in the yearbook club and it has 5 officers: 3 boys and 2 girls.  You all want to propose a big change to the yearbook so you want two officers to speak to Ms. Langan.  You want this committee to have one boy and one girl.  How many different choices of committees can you make?

Make a Tree Diagram Choices of Officers: since there are 3 boys and two girls to choose from, let’s identify the boys as B1, B2, and B3, and the girls as G1 and G2. B1B2B3 G1G2 G1G2 G1G2 How many combinations are there? Just count the number of arrows! 6 Ways!

Shortcut?  So if you count the ending outcomes, there were 6 choices of pairs that could talk to Ms. Langan.  How could we get this number without making a diagram? Is there a shortcut?  Let’s try another example to find out.

Lindsey’s Closet  Lindsey is going to be late because she can’t decide what to wear!  She has a choice of 4 shirts, 3 pairs of jeans, and 2 pairs of shoes.  Let’s make a tree diagram to represent this.

Put the largest # of choices on top Name some choices of outfits. Follow the choices down from the shirt to the shoes. How many choices are there all together? – Count the bottoms.

Let’s make a generalization…  3 boys and 2 girls = 6 combinations  4 shirts, 3 jeans, 2 shoes = 24 combinations THE COUNTING PRINCIPAL SAYS …  If one event can occur in “m” ways and another event can occur in “n” ways, then then MULTIPLY ALL THE CHOICES TOGETHER to discover the total number of combinations!

Let’s Go Outback Tonight…  Ms. Bright’s sister works at Outback.  When I go there, I have a choice of 4 different steaks, soup or salad, and 5 sides (garlic mashed potatoes, french fries, baked potato, vegetables, or rice)  How many different meals can I make?  4 x 2 x 5 = 40 different steak meals