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| Video 5: Probability |
| Tree Diagrams |
Overview
Students review the basic concepts of probability and move on
to complementary events, independent and dependent events and
compound events. They then create tree diagrams showing the sample
space or the number of possible outcomes.
Objective
Students will be able to write the probability and complement
of an event. They will be able to tell if the event is dependent
or independent.
Standards
Addressed
Mathematics — Data Analysis
Grade 4
Probability, Benchmark G
13. List and count all possible combinations
using one member from each of several sets, each containing
two or three members;
e.g., the number of possible outfits from three shirts, two
pairs of shorts and two pairs of shoes.
Grade 5
Probability, Benchmark H
07. List and explain all possible outcomes
in a given situation.
Probability,
Benchmark I
08. Identify the probability of events
within a simple experiment, such as three chances out of eight.
09. Use 0,1 and ratios between
0 and 1 to represent the probability of outcomes for an event,
and associate the ratio with the likelihood
of the outcome.
Grade 7
Probability, Benchmark I
07. Compute probabilities of compound
events; e.g., multiple coin tosses or multiple rolls of number
cubes, using such methods
as organized lists, tree diagrams and area models.
Grade 8
Probability, Benchmark J
11. Demonstrate an understanding that
the probability of either of two disjoint events occurring
can be found by adding the probabilities
for each and that the probability of one independent event
following another can be found by multiplying the probabilities.
Procedure
- On the board, draw this circle:
- Ask the students to take out
a piece of paper and number it from one to five. Have them answer
using correct symbolism.
If you were to spin this spinner, determine
the following:
What is the probability that you would
land on the green section? P(green) = 1⁄5
What is
the probability that you would land on blue or green? P(
blue or green) = 2⁄5
What is the probability you would
land on red? P(red) = 0⁄5
(an impossibility)
What is the probability that
you would not land on brown? P(not brown) = 4⁄5
What
is the probability that you would get the blue, brown,
purple, orange or green sections? P(blue,
brown, purple,
orange or green)
= 5⁄5 or 1 (a certainty)
- Have the students exchange
and correct each others’ papers.
- Discuss the following
concepts:
Complementary events are ones that are equal to
one. For examples, the probability of orange is one-fifth
and the
probability of
not orange is four-fifths. These are complementary
because added together they equal one.
Independent events are
when two events have nothing to do with each other. For example: “It’s
raining and I’m
going to the movie.”
Dependent events are when
the occurrence of one event depends upon the other.
For example: “If
it rains, I’m going to the movie.”
- Introduce the
concept of compound events as two or more independent events
that have a single outcome.
- Ask the students to list all
of the possible outcomes of getting heads-up when tossing a
coin and getting a five
when rolling
a die.
- Have them label this list the sample space.
- Ask the students
if they can think of a way to graph this.
- Introduce tree diagrams
using the following example.
Students can get either heads
or tails when they toss a coin (two outcomes on the
first event). They
can get a
one, two,
three, four, five or six when they throw
a die (six outcomes on the
second event). To show all of the possible
outcomes, ask the students to make a tree diagram with
you and list the
sample
space.
- Have the students make a tree diagram of
the following problem.
You want an ice cream cone. You have
a choice of a waffle cone or a regular cone. You have
the choice
of chocolate,
vanilla
or butter pecan ice cream. How many
different ice cream
cones could you have? List the sample space.
- Give
the student the following example with three events:
You are
going to a ball game. You have shorts or jeans; a blue,
yellow or pink T-shirt;
and a cap
or scarf
to wear. How many
different outfits could you have?
List the sample space.
- Ask the students how they could
find out the number of items in the sample space.
Make a
table of the
three problems they
just did.
| Event 1 |
Event 2 |
Event 3 |
# in Sample Space |
| 2 (coin sides) |
6 (1, 2, 3, 4, 5, 6) |
|
12 |
| 2 (cone types) |
3 (ice cream flavors) |
|
6 |
| 2 (pants) |
3 (T-shirts) |
2 (outerwear) |
12 |
Students should understand that when
you have compound events, all you
need to do
to find
out how many
events are in the
sample space is to multiply the number
in each related event together.
- Assign the
following problem for homework or for assessment:
Maria has
one green, one pink, one red and one yellow shirt.
She also has one
pair each
of black
jeans
and white jeans.
Make a tree diagram. List the
sample space. Tell how many events there
are.
Evaluation
Give the students the following problem:
Jean and Joe wanted
to join the band. They could pick marching band or concert
band. The instruments available to them
were flute, trombone, drums or oboe. How many possible
options could they have? Draw a tree diagram and list the sample
space.
| Sample space correct |
10 points |
| Tree diagram correct |
10 points |
| Correct number of events |
5 points |
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