11-56th grade math

Counting Methods

Objective• To use a tree diagram or the counting principle to

find the total number of outcomes for an event.

• Why? To know how find a different way of finding possible outcomes CHOICES. You can use what you know from older lessons: Make a List, Make a Table, Draw a Diagram to help you make tree diagrams or the counting principle.

California State Standards SDP 3.1 : Represent all possible outcomes of

compound events in an organized way (e.g., … grids, tree diagram) and express the theoretical probability of each outcome.

SDP 3.3 : Represent probabilities as ratios, … and percentages between 0 and 100 …

Vocabulary• Tree diagram

– A diagram used to organize outcomes of an experiment to make them easier to count. It sometimes does not matter which choice is addressed first, so more than one tree diagram can be correct. Count last row for outcomes.• Different outfits can be made w/black, navy, or green pants with a

white or yellow shirt.

• Counting principle– If one choice can be made in m ways and a second choice can be

made in n ways, then the two choices can be made together by m x n ways.• m = pants; n = shirts• m x n = pants x shirts = 3 x 2 = 6

• Counting grid– Use a grid to represent the different ways of sorting out the choices.

PantsShirts

Black (B)

Navy (N)

Green (G)

Yellow (Y) BY NY GY

White (W)

BW NW GW

How to Use the Tree Diagram

1) Read the problem2) Decide what choice will

be first3) Branch the second

choices off the first.4) Check your work

Make sandwich:Bread- pita, tortillaFilling- chicken, beef, veg.

6 ways to make a sandwich

How to Use the Counting Principle 1) Read the problem2) Decide what choice will

be first to be m3) The second choice will

be n.4) Multiply m x n5) The product is the

number of outcomes possible.

Make sandwich:Bread- pita, tortillaFilling- chicken, beef, veg.

Bread = mFilling = nBread x filling =2 x 3 = 66 outcomes of making

sandwiches

How to Use the Counting Grid1)Read the problem2) Decide what choice will be

first to be top row of the grid. Separate each choice into a column

3) The other choice will be separated into a row

4) ‘Multiply’ or put together the column and row of choices

5) The number of squares in the grid is the number of outcomes possible.

Make sandwich:Bread- pita, tortillaFilling- chicken, beef, veg.

6 squares = 6 outcomes for making sandwiches

BreadFilling

Pita Tortilla

Chicken PC TC

Beef PB TB

Veg. PV TV

How to Write the Probability of a Ratio1)Read the problem2) Decide the numerator as

the number of favorable outcomes (asking).

3) Write the denominator as the total number of possible outcomes.

4) The number of squares in the grid is the number of outcomes possible.

P(beef)Bread- pita, tortillaFilling- chicken, beef, veg.

Bread x filling = 2 x 3 = 66 outcomes of making sandwiches

Sandwiches w/beef= Bread x beef = 2 x 1 = 2

Ratio = 2 = 1 6 3

1/3 = .33 = 33%

Try It!Draw a tree diagram from

the chart below to show choices of sauce and pasta.

6 different ways to make a dish of noodles

Sauce Pasta

Alfredo Spaghetti

Vegetable Fettuccine

Meat

Try Some More!

2) P (spaghetti with veg. sauce)

3) P (fettuccini)

2) P (spaghetti) = 1P (veg.) = 11 x 1 = 1Total outcomes = 61 ≈ 0.167 ≈ 16.7% or ≈ 17% 6

3) P (fettuccini) = 3Total outcomes = 63 = 1 = 0.5 = 50%6 2

Try a bit more…

Use the counting principle to find the number of possible outcomes, taking one from each category.

4) 5 vegetables, 7 fruits5) 12 colors, 4 posters, 3

sizes

4) 5 x 7 = 35 outcomes

5) 12 x 4 x 3 = 144 outcomes

Objective Review • To use a tree diagram or the

counting principle to find the total number of outcomes for an event.

• Why? You now know how find a different way of finding possible outcomes CHOICES. You can use what you learned from older lessons: Make a List, Make a Table, Draw a Diagram to help you make tree diagrams or the counting principle.

• If there are m possible outcomes for the first event and n outcomes for the second event, then there are m x n total possible outcomes.

Independent Practice

• Complete problems 6-15

• Copy original problem first.

• Show all work!

• If time, complete Mixed Review: 16-20

• If still more time, work on Accelerated Math.