Middle school maths, year 2 · unit 9
Counting and Probability
Whether it rains tomorrow, what the die will show: you cannot know before it happens. But how readily it happens can be measured with a number. It begins with counting everything that could happen, missing nothing. We start by counting the faces of a die.
Count them one by one, missing none
A board to arrangeRoll a die and there are six faces it can show. Drag the faces that fit the condition onto the tray and count them.
One of each: multiply
Joining boardIn how many ways can you pick one top and one bottom to wear? Draw lines from the tops' dots to the bottoms' dots.
When order matters, and when it does not
A board to arrangeLining three people up, choosing a president and a vice-president, choosing two representatives. Are the counts the same? Drag the people into the boxes.
Probability: how much of the whole
A board to arrangeDraw one marble from the tray without looking: the chance of red is written as a number. Drag marbles in and out to change that number.
Toss many times and it shows
A board to adjustWhat does it mean to say a coin has probability 1/2 of landing heads? Toss it yourself. The results differ every time.
See it whole, as a table
A board to adjustThrow two things together and the cases multiply. Lay every case out as a cell, then drag the handle to light the cells.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- A result that can come out when you try is an event, and the number of ways it can happen is the number of outcomes. Miss nothing and count nothing twice.
- For two events that cannot happen together, the ways that one or the other happens are added (or). When one event follows for each way of another, the counts are multiplied (and).
- To count line-ups, multiply seat by seat from the front: 3 people give 3 × 2 × 1 = 6. If swapping places gives the same case, divide by the number of times it was counted: 2 representatives from 3 is 3 × 2 ÷ 2 = 3.
- When every case is equally likely, probability = (ways the event can happen) ÷ (all the ways). Repeat the same experiment many times and the relative frequency draws near this value.
- A probability is a number from 0 to 1: 1 if the event must happen, 0 if it cannot. If it happens with probability p, it fails to happen with probability 1 − p.
- For two events that cannot happen together, add to get the probability of one or the other; for two events that do not affect each other, multiply to get the probability of both. When in doubt, lay every case out in a table and count.