Primary mathematics/Probability

Children and Probability
In mathematics probability tells us how likely something is to happen.

If something is absolutely certain to happen, we say its probability is one. If something cannot possibly happen, we say its probability is zero. A probability between zero and one means we don't know for sure what will happen, but the higher the number the more likely something is to happen. If something has a probability of 0.25, most people would say "it probably won't happen".

Unfortunately, children have a hard time understanding what a probability of 0.25 suggests. Children are much more adept at understanding probability in terms of percentages (25%), and fractions (i.e., "It will happen about ¼ of the time"), so it is very important that connections between probability and the development of those skills and understandings occur together.

Models
Probability is best taught with models. Two of the most effective and commonly used models used to teach probability are area and tree diagrams

Area diagrams give students an idea of how likely something is by virtue of the amount of space reserved for that probability. Consider the toss of a coin:

Students can imagine that every time they toss a coin, they need to put it into the appropriate side of the rectangle. As the number of throws increases, the amount of area needed to enclose the coins will be the same on both sides. The same probability can be shown using a tree diagram:



In this simple tree diagram, each line (or event path) will be followed ½ of the time. Note that both diagrams show all of the possibilities, not just the probability of one event.

Both of these models can become more complex. Consider all of the possibilities of throwing two coins in series:

In this area rectangle the instances where heads has been tossed twice in a row is represented in the upper left hand corner. Note that whereas the probability of tossing both heads and tails (in any order) is 50% or 1/2, and the probability of tossing two heads in a row is 25% or 1/4.



While the tree diagram gives us the same information, the student needs to understand that all of the events at the bottom have an equal chance of occurring (This is more visually evident in an area diagram). However, the tree diagram has the advantage of better showing the chronology of events (in this case the order of events is represented by downward motion).

While it may appear that both of these models have strengths and weaknesses, that is not the point. By becoming familiar with these and other models, students gain a more robust and diverse understanding of the nature of probability.

Consider the following example problem:

If at any time a Dog is just as likely to give birth to a male puppy as she is to give birth to a female puppy, and she has a litter of 5 puppies, what is the probability that all of the puppies will be of the same sex? While this problem can be easily answered with the a standard formula, $$P = 2 \left ( \frac{1}{2} \right ) ^n$$, where n is the number of puppies born, this nomenclature is not only unintelligible to students, but teaching it to students gives them no understanding of the underlying concepts. On the other hand, if they have explored problems similar to this with a tree diagram, they should fairly easily be able to see patterns emerging from which they can conjecture the "formula" that will find the correct answer. Consider that a student has drawn a tree diagram that shows the permutations with three puppies in the litter:

If a teacher was to have this student share their work, the class of students in looking at this model would be able to see that there are 8 possible outcomes. They should also notice that the only instances where litters of all males or all females can be found are on the far ends of the tree.

Students are always encouraged to look for patterns in many of their mathematical explorations. Here, students start to notice that the number of possibilities can be determined by multiplying the number 2 by itself exactly the number of times there is to be a new outcome (birth). In this case there are 3 outcomes (puppies born) in a row. So 2 x 2 x 2 = 23, or 8 possible outcomes. Once students start to make this connection from seeing this pattern, they may conjecture that if there are 5 puppies in a litter, the possible number of outcomes is 25, or 32, of which only 2 (represented by the combinations MMMMM, and FFFFF on the extreme ends) will result in a litter puppies of all the same sex.

Teachers should always be looking for students to make connections with other skills they are currently learning. In this case the teacher might ask the students to state the answer as a reduced fraction (1/16) and a percentage (6.25%).