Learning Math: Data Analysis, Statistics, and Probability
Probability Part C: Analyzing Binomial Probabilities (45 minutes)
In This Part: Making a Tree Diagram
A tree diagram is a helpful tool for determining theoretical or mathematical probabilities. Let’s begin by examining the problem of tossing a fair coin. We’ll focus on the number of heads that occur in a certain number of tosses. See
A tree diagram for the toss of a single coin has two branches that represent the two possible outcomes of this random experiment. In this tree diagram, the red branch represents the outcome “heads” (H), and the blue branch represents the outcome “tails” (T):
For a single toss, the outcome is either heads or tails. Since we’re looking at the number of heads that occur, the possible values from one toss are either 1 (heads) or 0 (tails).
We can extend the tree diagram to show more than one coin toss. Use the Interactive Activity to see how we construct the diagram. Try several rounds of two, three, and four tosses, and record your outcomes.
Let’s expand our tree diagram to two tosses of a fair coin. Again, each red branch represents the result heads, and each blue branch represents the result tails.
The tree diagrams below illustrate the four possible paths along the branches when you toss a coin twice:
Path 1: First Toss — Heads, Second Toss — Heads (Abbreviated HH):
Path 2: First Toss — Heads, Second Toss — Tails (Abbreviated HT):
Path 3: First Toss — Tails, Second Toss — Heads (Abbreviated TH):
Path 4: First Toss — Tails, Second Toss — Tails (Abbreviated TT):
In this video segment, Professor Kader demonstrates how to construct a tree diagram. As you watch, ask yourself, What does a path on a tree diagram represent? View this segment after you’ve completed the Interactive Activity.
Note: In the experiment conducted by the onscreen participants, participants tried to guess whether dice would land on an even or an odd number. If their guess was correct, the outcome was labeled “C”; if incorrect, the outcome was labeled “I.”
In This Part: Probability Tables
If we assume that the coin is fair, each outcome (heads or tails) of a single toss is equally likely. This probability table summarizes the mathematical probability for the number of heads resulting from one toss of a fair coin:
Let’s take a closer look at the tree diagram for two coin tosses. Each red branch represents the result heads (or H). Each blue branch represents the result tails (or T). The outcome associated with each path is indicated at the end of the path, together with the number of heads in that outcome.
Since we are tossing a fair coin, each of the four outcomes (HH, HT, TH, TT) is equally likely. See Note 8 below.
Use this tree diagram to explain why the likelihood of getting exactly one head in two coin tosses is not the same as the likelihood of getting zero heads in two coin tosses.
What is the probability of each possible outcome? The possible values for the number of heads from two tosses are two (HH), one (HT, TH), or zero (TT).
This probability table summarizes the mathematical probabilities for the number of heads resulting from two tosses of a fair coin:
The purpose of Part C is to investigate the idea and use of the binomial probability model.
Part C analyzes the question of how many times you will get heads when you toss a coin a given number of times. The outcome of a coin toss is a random event, so any answer to the question requires the use of probability.
Tree diagrams are used to analyze the coin toss problem. A tree diagram is a useful tool for analysis and also an effective pedagogical device.
The tree diagram for two tosses of a fair coin helps to relate the four possible outcomes to the number of heads for each outcome:
The key idea is to determine how many outcomes (paths through the tree) contain zero heads, one head, or two heads, respectively.
Some people have great difficulty with the notion that some values occur more often than others. They think that if there are three possible values (zero, one, and two), then each must be equally likely. The tree diagram may clarify the different frequencies of occurrence for different possible outcomes.
Two branches of the tree end with one head out of two tosses (HT and TH), and only one branch ends with zero heads (TT). Therefore, it is more likely to get one head than no heads.
Here is the tree diagram for three tosses of a fair coin:
Here is the completed probability table:
Session 1 Statistics As Problem Solving
Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced.
Session 2 Data Organization and Representation
Explore different ways of representing, analyzing, and interpreting data, including line plots, frequency tables, cumulative and relative frequency tables, and bar graphs. Learn how to use intervals to describe variation in data. Learn how to determine and understand the median.
Session 3 Describing Distributions
Continue learning about organizing and grouping data in different graphs and tables. Learn how to analyze and interpret variation in data by using stem and leaf plots and histograms. Learn about relative and cumulative frequency.
Session 4 Min, Max and the Five-Number Summary
Investigate various approaches for summarizing variation in data, and learn how dividing data into groups can help provide other types of answers to statistical questions. Understand numerical and graphic representations of the minimum, the maximum, the median, and quartiles. Learn how to create a box plot.
Session 5 Variation About the Mean
Explore the concept of the mean and how variation in data can be described relative to the mean. Concepts include fair and unfair allocations, and how to measure variation about the mean.
Session 6 Designing Experiments
Examine how to collect and compare data from observational and experimental studies, and learn how to set up your own experimental studies.
Session 7 Bivariate Data and Analysis
Analyze bivariate data and understand the concepts of association and co-variation between two quantitative variables. Explore scatter plots, the least squares line, and modeling linear relationships.
Session 8 Probability
Investigate some basic concepts of probability and the relationship between statistics and probability. Learn about random events, games of chance, mathematical and experimental probability, tree diagrams, and the binomial probability model.
Session 9 Random Sampling and Estimation
Learn how to select a random sample and use it to estimate characteristics of an entire population. Learn how to describe variation in estimates, and the effect of sample size on an estimate's accuracy.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in this course can be applied through a case study of a K-2 teacher, Ellen Sabanosh, a former course participant who has adapted her new knowledge to her classroom.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in this course can be applied through case studies of a grade 3-5 teacher, Suzanne L'Esperance and grade 6-8 teacher, Paul Snowden, both former course participants who have adapted their new knowledge to their classrooms.