Which bag would you choose?
Open the full chapter tools, quiz and word problems
Use these for more practice or to work alongside the textbook.
1Probability is a measurement
Length is measured in metres. Probability measures how likely an event is. The scale runs from 0 to 1. A probability of 0 means the event cannot happen. A probability of 1 means it is sure to happen. Everything else sits in between, and the closer to 1, the more likely.
The textbook's first picture is a deck of six cards, some purple and some green, and asks how likely it is that a card picked at random is purple. Tap the cards to change their colours and watch where the marker lands.
Impossible
Even chance
Certain
Put words to the numbers
Before any calculation, the chapter wants you to be able to say roughly where an event sits on the scale. Pick a label for each event. Some of these come from Exercise 7.1, the rest from the table in section 7.1.2.
2Randomness, and measuring it by doing
A coin toss is random. You know the two things that can happen, heads or tails, but you cannot say which one will come up this time. A single toss tells you almost nothing about the coin. Many tosses do. Counting how often something happened over many tries gives the experimental probability, which the book also calls the relative frequency.
Toss a few times. With 10 tosses the line jumps around. Press ×1000 a few times and watch it settle near the dashed 0.5 line.
The same thing with a die
A fair die has six faces, so each face should come up about one sixth of the time. Roll and compare the counts with the dashed line, which marks where each bar would be if every face came up exactly as often.
Worked example from the book: roll a die 50 times and suppose a 4 comes up 8 times. The experimental probability of a 4 is 8/50 = 0.16, or 16%. The theoretical probability is 1/6 ≈ 0.167. They are close but not equal, and nobody expects them to be equal after only 50 rolls.
3Theoretical probability: counting instead of doing
When every outcome is equally likely, you do not need an experiment at all. You count the outcomes you want and divide by the number of outcomes there are. The book writes the probability of an event as P(event).
"Favourable" only means "the ones the question is about". Example 4 in the book picks a letter at random from PROBABILITY and asks for the chance of a B. Two of the eleven letters are B, so P(B) = 2/11 ≈ 0.182. Try other words and letters. The word PEACE from the end-of-chapter exercise is a good one.
4Using data from a sample
Example 5 in the book asks 50 students in one class for their favourite fruit and gets: 20 mango, 15 apple, 10 banana, 5 grapes. Pick one of those students at random and the chance their favourite is mango is 20/50 = 0.4. The school has 1500 students. Nobody is going to ask all of them, so the class is used as a sample and the whole school is the population. If 40% of the sample likes mango, a reasonable guess is that about 40% of the school does too, so buy about 600 mangoes.
Move the sliders to change the survey and the school size. The estimate for each fruit is the sample probability multiplied by the population.
A bigger sample gives a steadier estimate, and a sample that mixes students from different classes is better than one class. The book calls this sampling, and leaves the details of sample size and representativeness for later.
5The coin has no memory
Suppose a fair coin lands heads six times in a row. Many people feel that tails is now "due". It is not. The coin does not know what happened before. On the next toss, P(tails) is still 1/2. The book calls the mistaken feeling the Gambler's Fallacy, and gives the Snakes and Ladders version: three sixes in a row do not make a fourth six any less likely. Each roll is an independent event.
Here is a test. The computer tosses a coin over and over. Every time it sees a run of the chosen length, it writes down what the very next toss was. If the fallacy were true, the next toss would lean toward the other side.
Run it several times. The tails share stays near 0.5 whatever run length you choose. Long runs are rarer, so the computer needs more tosses to find 500 of them, but what follows a run is still a plain 50:50 toss.
6Sample spaces and events
The sample space, written S, is the list of every possible outcome of an experiment. Two rules: it must include every outcome, and no outcome appears twice. The number of outcomes is written n(S). An event is the group of outcomes that fit your question. For a die, “an even number” picks out {2, 4, 6} from {1, 2, 3, 4, 5, 6}. The textbook calls that group a subset of the sample space. The event's probability is n(E) divided by n(S), as long as all outcomes in S are equally likely.
Choose an experiment, then tap outcomes to put them in the event. The presets are events from the chapter.
7Tree diagrams for two-step experiments
Tossing a coin twice, or drawing a ball and then another ball, is a multi-step experiment. A tree diagram draws each step as a set of branches. Every path from the root to a leaf is one complete outcome, and the probability written on each branch multiplies along the path. Drawing the tree is a reliable way to check that the sample space is complete.
What matters is whether the first item goes back before the second draw. If it goes back, the second step looks exactly like the first. If it is kept aside, there is one fewer item, and the second-step fractions change.
8Two dice: 36 outcomes
Rolling two dice gives 36 equally likely outcomes, one for each pair (first die, second die). The sums run from 2 to 12, but the sums are not equally likely. A 7 can be made six ways and a 2 only one way. The grid shows every outcome with its sum. Pick a condition to see which cells count.
9Probability from area
The last end-of-chapter problem drops a dye at random onto a 3 m by 2 m rectangle with a circle of diameter 1 m drawn on it. Here "at random" means every point is equally likely, so the probability of landing inside the circle is the circle's area divided by the rectangle's area. Drop dots and compare the share that land inside with that ratio.
Circle area = π × 0.5² ≈ 0.785 m². Rectangle area = 6 m². Ratio = 0.785 / 6 ≈ 0.131. The dots are drawn only up to 3000 to keep the picture readable, but the counts keep going.
Quiz
Fourteen questions about the ideas rather than the arithmetic. Each answer shows the reason straight away. Your best score is remembered on this device.
Word problems from the chapter
These follow the exercise sets and the end-of-chapter questions, reworded. Type an answer as a fraction like 3/8, a decimal like 0.375, or a percentage like 37.5%, then press Check. A hint and the full working are there if you get stuck. Starred questions in the book are the harder ones and are marked here too.