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 cool connection between probability and algebra).

Do you want to know more? Yes or no?

*[PS – Aunt Athanasia used to say: In life you’ll have to make many many choices between paths. The right path will always be the most difficult one.]*

**1. LET’S TAKE CHANCES**, part 1

If you flip a coin, which is more likely? Heads (H) or tails (T)?

If you flip a coin twice, which is more likely? HH, HT or TT?

**In a Grade 1 classroom**

Grade 1 students used unplugged coding to simulate coin-tossing probability experiments.

This is the algorithm they followed for tossing 2 coins:

This is the paths diagram they used to keep track of results:

**The paths diagram**

The paths diagram is a conceptual tool.

The paths diagram helps young students understand why B is more likely.

Students can see that there are 2 paths leading to B, which makes B twice as likely as A or C.

**2. LET’S TAKE CHANCES, part 2**

If you flip a coin 5 times, one possible outcome is HHHHH. How likely is this outcome?

What other outcomes are possible? Which one is most likely?

**In a Grade 1 classroom**

When asked which might be more likely (A, B, C, D, E or F?) of we tossed a coin 5 times, Grade 1 students shared that A and F are the least likely, because they each had only one path leading to them.

**Sums of paths**

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

Notice a pattern?

- 1 + 1 = 2
- 1 + 2 + 1 = 4
- 1 + 3 + 3 + 1 = 8
- 1 + 4 + 6 + 4 + 1 = ?
- 1 + 5 + 10 + 10 + 5 + 1 = ?

**Calculating theoretical probability**

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?

**3. ALL POSSIBLE OUTCOMES**

Each additional coin flip doubles the possible outcomes?

Is this true? Can you prove it?

**4. PROBABILITY & ALGEBRA**

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

**5. 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.

### 6. TOSSING 2 COINS WITH CODE

The Scratch code shown on the right simulates tossing of 2 coins.

You may access, run and edit this code at https://scratch.mit.edu/projects/402742403/editor

- H = 0 and T = 1
- SUM = H + T
- if SUM = 0, we must have HH, and a green cat is stamped
- if SUM = 2, we must have TT, and an orange cat is stamped
- if SUM = 1, we must have HT, and a blue cat is stamped
- sample output is shown below