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