# 5. BINARY CHOICE

“I believe that we do not know anything for certain, but everything probably.”
― Christiaan Huygens (mathematician & physicist)

#### IN THE CLASSROOM

###### Ontario Curriculum
• Gr. 1-8
• D2 – describe the likelihood that events will happen, and use that information to make predictions
• D2.1 – describe the likelihood of events happening and use that likelihood to make predictions and informed decisions
• C1.3 – determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in patterns
• C3.1 – solve problems and create computational representations of mathematical situations
• Gr.5-8
• D2.2 – determine and compare theoretical and experimental probability
• Gr. 9
• B1.1 – research an algebraic concept to tell a story
• B2.2 – analyse, through the use of patterning, the relationships between the exponents of powers and the operations with powers, and use these relationships to simplify numeric and algebraic expressions
• C2 – apply coding skills to represent mathematical concepts and relationships dynamically, and to solve problems, in algebra and across the other strands
• Gr. 11-12
• the Binomial Theorem
###### Implementation
• Gr. 1-3
• Students conduct experiments to determine the likelihood of outcomes when tossing a coin once and twice.
• Using a tree diagram they explain that getting a H and T combination is twice as likely (there are 2 paths to destination B).
• Gr. 4-8
• Students also investigate connections to Yang Hui’s (aka Pascal’s) triangle.
• They also use code to model coin flips.
• Gr. 9-12
• Students also investigate connections to expansions of binomial powers, & the Binomial Theorem.

#### 5a. IN GRADES 1-3 CLASSROOMS

###### Tossing 1 coin

Q1. Working in pairs, students place a red/yellow (heads/tails) plastic counter (or a coin) in a cup.

• they cover the cup with one hand, and shake the cup
• they look at the counter (or coin)
• they use this “algorithm” below to decide if the result on the path is A or B
• using a show of hands, each pair indicates whether their result is A or B
• the teacher records a tally of the class results
• the teacher comments: “B won! It has the most tallies.”
• then asks: “Which letter would you predict will win next time? A or B? Why?”

###### Tossing 1 coin twice

Q2. The teacher presents this new algorithm and the new diagram.

• Using pair/share, students discuss in pairs, and then share with the whole class, how this algorithm is different from the previous one, and how the new diagram would be used.
• The teacher models the algorithm by tossing a coin twice and following the path to one of the letters A, B or C.
• The teacher asks: “If we did this as a whole class, and tallied the results, which letter would win? A, B or C? Why?”

#### 5B. TOSSING A COIN 5 times

Q5. The teacher presents the following paths diagram.

The teacher asks: “How do we alter the algorithm (on the right), so we can get to the letters at the bottom of the path?”

The teacher then asks: “If we did this as a whole class, which letters would be more likely to win? Which would be more likely to lose? And, can you explain why?”

#### 5C. PUZZLE 1 – Tossing 1 coin with PYTHON

Q3. The code below models tossing a coin 100 times and counting the number of heads and tails. Enter and run the code at https://cscircles.cemc.uwaterloo.ca/console/

• How does the code work?

Q4. Alter the code to model:

• 20 trials and 10 trials
• calculate the percents of H and T
• 1000 trials and 10,000 trials
• calculate the percents of H and T
• what is the effect of doing more or fewer trials?

#### 5D. PUZZLE 2 – Tossing 2 coinS with PYTHON

Q4. The code below models tossing 2 coins 100 times and counting the number of times we get HH, HT and TT.

• Notice that the code is missing some components.

#### 5E. BINARY CHOICE

When you toss a coin, you are making a binary choice. A choice between 2 options: heads or tails.

A light switch also offers a binary choice: on or off.

Digital devices, like smartphones, make decisions through complex combinations of ON and OFF, or 1 and 0. This is why the binary number system is especially well suited to how they function.

The idea of binary choice is powerful, and can be seen in probability (tossing coins) and in algebra (x + y), as well in the binomial theorem, which is a really powerful connection between probability and algebra.

Q6. Do you want to know more? Yes or No?

#### 5F. MATHEMATICAL SURPRISE

The first activity, where students toss a coin once, and where the outcomes of A and B are equally likely, sets the stage for young students to expect that outcomes A, B and C will also be equally likely when tossing a coing twice.

This leads to a mathematical surprise, as it turns out in their experiments that bB is more likely than A or than B. As well, the paths diagram allows students to make sense of the situation. leading to the conceptual insight that there are more paths that lead to B.

For higher grades that are also these surprises and insights:

• the patterns in Yang Hui’s triangle represent the sums of the paths leading to each intersection, and also represent the possible outcomes when tossing a coin repeatedly
• the possible outcomes represent the numeric coefficients of binomial expansions: HH + 2HT + TT is equivalent to x2 + 2xy + y2
• two seemingly disparate topics, probability of tossing a coin and the algebra of binomial expansions, can overlap due to the common underlying idea of binary choice: H or T and x or y

#### 5G. Sums of paths

Q8. The diagram on the right shows the number of paths leading to each node.

• Notice a pattern?
• What are the missing numbers below?
• 1 + 1 = 2
• 1 + 2 + 1 = 4
• 1 + 3 + 3 + 1 = 8
• 1 + 4 + 6 + 4 + 1 = ?
• 1 + 5 + 10 + 10 + 5 + 1 = ?

#### 5H. Calculating theoretical probability

Q9. The theoretical probability of getting 2 heads and 1 tail (HHT) tossing 3 coins is 3/8.

• Can you see how 3/8 can be derived from the information shown above?
• What is the theoretical probability of each of the following?
• HHHT when tossing 4 coins
• HHHTT when tossing 5 coins
• HHTTTT when tossing 6 coins?

#### 5I. Statistician Interview

Interview with Dr. Bethany White, University of Toronto.

###### Tossing a coin twice

Bethany White discusses coin tossing (and computational thinking) activities explored by Grade 1 students.

Q10. Let’s toss a coin to decide whether we walk left or right.

• If we toss the coin twice, where will we end up?
• Which path is more likely?

###### Tossing a coin 5 times

Q11. Let’s toss a coin 5 times.

• Which path are we more likely to follow?
• And what does this have to do with Pascal’s Triangle?

###### All possible outcomes

Q12. Let’s look at all the possible outcomes.

• What pattern do we notice?

###### Probability & algebra

Q13. There is a link between probability and algebra!

If you imagine H and T as X and Y, you may discover a powerful link between probability, algebra and Pascal’s triangle.

###### Weighted outcomes

If you plot the frequency of one of two equally likely events as a graph, its plot looks like a “bell”.

If the events are not equally likely, the graph skews to one side of the other.

#### 5J. PUZZLE 3 -Modelling OUTCOMES with SCRATCH

Q14. The Scratch code below simulates tossing of 2 coins. You may access, run and edit this code at https://scratch.mit.edu/projects/402742403/editor

• What does the code do?
• How does it do it?
• Alter the code to also show the percents of HH, HT and TT.