Download Required Knowledge Module Goals Helping with Homework and

Dear Family,
This module of your student’s Integrated Geometry course is called What Are My Child’s Chances? In this module, students examine various
probability models and their applications to the study of genetics. They use probability to solve problems involving genetic traits and disorders.
Required Knowledge
To be successful in this module, students will need certain concepts
and skills that they learned in previous math classes. For What Are
My Child’s Chances?, your student should already know:
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how to determine sample spaces.
how to make tree diagrams.
how to use fundamental counting principle.
how to to draw Venn diagrams.
how to calculate expected value.
how to evaluate exponential expressions.
Module Goals
During this module, your student will:
• identify complementary events.
• use Venn diagrams to determine sample spaces and theoretical
probabilities.
• collect theoretical and experimental probabilities.
• use Punnett squares to determine sample spaces and theoretical
probabilities.
• identify independent, dependent, and mutually exclusive events.
• investigate formulas for determining the theoretical probability of
two or more events, including P(A or B) and P(A and B)
• use tree diagrams to determine sample spaces and theoretical
probabilities.
Helping with Homework and Activities in this Module
Activity Activity Activity Activity
The goal of SIMMS Integrated Mathematics is to develop strong math skills through modeling, practicing, writing, and technology. You can help with
homework and encourage good math habits by asking questions like the suggestions below.
1. Which of the five traits in Table 2-1 on page 31 do our family members have?
2. Looking at Figure 2-1 on page 31, what does this wheel represent?
3. In class, how did you find the experimental probability of a person with free earlobes?
1. In class, how did you use the Punnett square to find the possible gene combinations?
2. What is the difference between finding the experimental probability and theoretical probability of an event?
1. What are two events that are dependent to each other? Why are they dependent?
1. How is drawing a tree diagram helpful in determining probability?
2. When flipping one coin, there are two possibilities: heads or tails. Why are these two events mutually exclusive?
In your student’s math notebook, you should find Explorations completed from each Activity, along with completed examples from problems done in
class, notes on the math concepts in the module, and descriptions of the vocabulary words.
Having Conversations about the Mathematics in this Module
There are several ways you can help your student be successful in this specific module.
• Have your student share their math notebook with you. Ask them to show you what they have recorded about the
activities for What Are My Child’s Chances?. Ask your student to explain the important concepts in these activities.
• What is one thing you have learned about inheriting genetic traits and probability?
• How would you create a tree diagram?
• How does Venn diagram help you find probabilities? (see page 33)
• Look over your student’s homework and make sure all questions are answered and that explanations are clear.
Some of the important mathematical concepts your student is learning in What Are My Child’s Chances? are provided on the back of this page. If
you have any questions or concerns about this module or your student’s progress in this course, please feel free to call or e-mail me.
Sincerely,
Important Concepts
The sample space is the set of all possible
outcomes for an experiment.
The experimental probability of an event equals
the number of times an event occurs divided by the
total number of trials.
Examples and Pictures
For example, the sample space for rolling a number cube with its sides labeled 1 through 6
is {1, 2, 3, 4, 5, 6}. In all, there are 6 possible events.
For example, there are six possible outcomes from rolling a number cube with its sides
labeled #1-6. Out of 100 trials, the #6 may come up 36 times. Therefore, the probability is
equal to 36 chances out of 100. 36 = .36 = 36% This may not be the true probability, but
100
as the number of trials increases the experimental probability becomes more accurate.
The theoretical probability of an event equals the
number of outcomes in the event divided by the
total number of outcomes in the sample space.
A Venn diagram is a way of showing the
relationship between sets. The numbers or objects
in each set are placed in circles. If the sets have
elements in common, the circles are overlapped
and the common elements are placed in the area
formed by the intersection.
The complement of a set is the outcomes not in an
event. For example, the complement of Even
numbers #1-20 would be all the number outside of
the circle {1,3,5,7,9,11,13,15,17,19}
For example, the theoretical probability of rolling a 2 when rolling a number cube with its
sides labeled 1 through 6 is equal to one chance out of six. 1 = .1666 = 16.67%
6
The following Venn Diagram compares two sets of numbers:
Even Numbers #1-20:{2,4,6,8,10,12,14,16,18,20}
Squared Numbers #1-20: {1,4,9,16}
All numbers #1-20
Note: The numbers
4 and 16 are in the
intersection of
the two circles
because
they are
in both sets.
1 9
Square numbers #1-20
Mother
F
f
A Punnett square is a chart used to identify the
possible combinations from a genetic cross
between a mother and father.
Father
F
FF
Ff
f
Ff
ff
4
16
2 6 8 10
12 14 18
20
3, 5, 7, 11,
13, 15, 17, 19
Even numbers #1-20
The probability of Free Earlobes is
3
.
4
The probability of Attached Earlobes is
1
.
4
This example shows the cross of the genetic trait “free ear lobes”. Capital F is the dominant
trait, f is the recessive trait. If one dominant trait is present then the dominant trait is passed
down, for example with FF, Ff. The only possibility for possessing the recessive trait is ff.
An independent event is an event not affected by
another event.
To find probability of independent events,
multiply the probability of event 1 by event 2.
Suppose you roll a number cube labeled 1 through 6 two times. Each roll of the number
cube is an independent event because the probability of rolling the number cube is the
same every time. Finding the probability of rolling two consecutive “1s”. Multiply the
probability of the first roll and the second roll.
1 1 1 or 0.027 or 2.7% chance of rolling two “1s”
• =
6 6
A dependent event is an event affected by another
event.
To find the probability of dependent events
(conditional probability) multiply the probability of
the first event by the probability of the second event
taking place after the first event.
Two events are mutually exclusive if they cannot
occur at the same time in a single trial.
36
Suppose a bag holds 5 red marbles and 5 black marbles. You draw one marble and set it to
the side. Then you draw a second marble and set it to the side. These two events are
dependent because the probability of drawing a red or black marble on the second drawing
has changed based upon the first drawing. To find the probability of drawing 2 red marbles,
multiply the probability of the first event and the probability of the second event
5 4 20
or 0.22 or 22% chance of drawing 2 reds.
• =
10 9
90
Consider the toss of a single coin. In this case, head and tails are mutually exclusive,
because both cannot occur at the same time on a single toss. Another example might be a
person could be both right-handed and have free earlobes. In this case, these two events
are not mutually exclusive.