Secondary 1 Probability Worksheet

10 of the 11 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 10 of the 11 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Modes of Representation and the Enumeration of Possible Outcomes

   A skate shop in Verdun sells a starter kit. The buyer picks one helmet colour — black, teal or white — and one size of knee pads — medium or large.

   1. Draw a tree diagram showing every possible kit.
   2. Write out the complete list of kits, using a pair such as (black, medium).
   3. How many different kits are possible?
2. Q2Probability

   A school raffle sells 40 tickets, numbered 1 to 40, and one winning ticket is drawn at random. Léa bought the 6 tickets numbered 12 to 17.

   1. Describe the random experiment, and say how many outcomes the sample space contains.
   2. Describe the event “Léa wins”, and say how many outcomes it contains.
   3. Find the probability that Léa wins. Give it as a fraction in lowest terms, as a decimal and as a percentage.
3. Q3Random Experiments Where Order Matters and Where Order Does Not Matter

   The four members of a robotics team are Anaïs, Bruno, Chloé and Dimitri.

   1. Two of them are chosen to carry the robot to the bus. Here the order of the choice does not matter. List all the possible pairs and count them.
   2. Two of them are chosen instead as captain and then as assistant captain. Here the order does matter. Count the possibilities.
   3. Which of the two counts is larger, and by how many times? Explain why.
4. Q4Random Experiments With One or More Steps

   A game has two steps. First you spin a spinner with three equal sectors numbered 1, 2 and 3; then you flip a fair coin.

   1. Build a table showing every possible result of the game.
   2. Find the probability of getting an even number and heads.
   3. Find the probability of getting an odd number.
5. Q5Random Experiments With and Without Replacement

   A bag holds four tokens marked 1, 2, 3 and 4. A token is drawn, then a second one; the two numbers are written down in the order they come out.

   1. How many results are possible if the first token is put back before the second draw? And if it is not put back?
   2. In each of the two versions, find the probability that the two numbers add up to 5.
6. Q6Sample Space

   A carnival game flips a fair coin and rolls a four-sided die numbered 1 to 4.

   1. Write the sample space of this experiment, using set braces and pairs such as $(\text{H},3)$.
   2. Event $E$ is “the die shows a number greater than 2”. List the outcomes of $E$ and find $P(E)$.
   3. Event $F$ is “the coin shows tails and the die shows 1”. Find $P(F)$.
7. Q7The Concept of ``Or'' and ``And'' in Probability

   Twenty cards numbered 1 to 20 are shuffled and one is drawn at random. Event $A$ is “the number is a multiple of 3” and event $B$ is “the number is a multiple of 5”.

   1. List the outcomes of $A$ and the outcomes of $B$.
   2. Find $P(A\text{ and }B)$.
   3. Find $P(A\text{ or }B)$.
8. Q8The Intersection and Union of Sets

   In a group of 30 students, 18 play hockey, 14 play soccer and 7 play both.

   1. Draw a Venn diagram with two overlapping circles and write the correct number in each of its four regions.
   2. How many students play neither sport?
   3. One student is chosen at random. Find the probability that the student plays exactly one of the two sports.
9. Q9The Types of Probability

   This question is built around a diagram or a table of values. Open it in the PDF.

10. Q10Types of Events

    A box holds ten tokens numbered 1 to 10, and one token is drawn at random. For each event below, give its probability and name the most precise type that applies, choosing from certain, impossible, elementary and probable:

    1. $A$: the number drawn is less than 11;
    2. $B$: the number drawn is 0;
    3. $C$: the number drawn is 7;
    4. $D$: the number drawn is even.

    Then say whether $D$ and the event $E$: “the number drawn is odd” are complementary, and justify your answer.

11. Q11Synthesis — drawing on several sheets in this topic

    A vending machine at the arena drops one bottle at random. The flavour is lime, cola or orange, all equally likely, and the size is small or large, both equally likely; the two choices do not affect each other.

    1. Represent the experiment with a table or a tree, and write the sample space.
    2. Find the probability of getting an orange bottle or a large bottle.
    3. Are “the bottle is small” and “the bottle is lime” compatible or incompatible? Justify.
    4. Name the event complementary to “the bottle is large”, and give its probability.

The 11 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

"Equally likely" is the condition the whole method rests on

Counting favourable outcomes and dividing by the total works only when every outcome in the list is as likely as every other. That clause is invisible when you are rolling a fair die, and it is the difference between an answer and a guess everywhere else. Q2 sets up the honest case — forty numbered raffle tickets, one drawn at random, six of them held by the same person — and asks you to describe the experiment, the sample space and the event in words before any fraction appears.

Writing those three descriptions out is not ceremony. It is where you find out whether your list is the right one, and it is what makes an answer defensible later on.

Representing a two-step experiment (Q1, Q4, Q6)

A tree and a table do the same job and suit different shapes of problem. A tree handles stages one after another, and Q1 builds one for a kit with a colour chosen at the first stage and a size at the second. A table is quicker when there are exactly two stages, since the outcomes are simply its cells — Q4's spinner and coin, Q6's coin and four-sided die.

Counting the outcomes then comes from the picture rather than from a formula: every branch of the first stage splits into the branches of the second, so the number of branch ends is the product of the stages. Three colours and two sizes give six kits; two coin faces and four die faces give eight outcomes. Draw the tree once and that multiplication stops being a rule to remember.

Order matters, or it does not (Q3)

Q3 gives four teammates and asks two questions about choosing two of them. Choosing a pair to carry equipment does not care about order — AB and BA are the same pair. Choosing a captain and then an assistant does care, because the two roles are different, so AB and BA are two different results.

The ordered count comes to exactly twice the unordered one, and the third part asks why. The reason is worth being able to say: every unordered pair produces exactly two ordered results, never three, because two things can only be arranged in two orders. That is the reasoning this topic wants — no formula, just an accurate account of what the list contains.

With replacement and without (Q5)

Q5 draws two tokens from four, one after the other, in both versions of the experiment. Putting the first token back leaves four choices for the second draw; not putting it back leaves three. So the sample space shrinks, and it shrinks in a specific way: the outcomes that disappear are exactly the ones that repeat a token.

That is what makes the second part interesting. The favourable outcomes — pairs adding to five — never repeat a token, so all of them survive. The same favourable count over a smaller sample space gives a larger probability, and being able to explain that in a sentence is worth more than the two fractions.

"And" narrows, "or" widens — and "or" double-counts (Q7, Q8)

"A and B" keeps only the outcomes in both lists; among the cards numbered 1 to 20 in Q7, exactly one number is both a multiple of 3 and a multiple of 5. "A or B" keeps the outcomes in either list — and here is the trap the topic is built around: if you count the two lists and add, any outcome sitting in both has been counted twice.

Q8 is the same arithmetic drawn as a Venn diagram, and the reliable way to fill one in is to start with the overlap and subtract outwards. Eighteen play hockey and seven play both, so eleven play hockey only. Then the four regions must add to the size of the group, which both completes the diagram and checks it.

Theoretical and experimental probability (Q9)

Two different questions, and both are legitimate. A theoretical probability comes from the structure of the experiment: four equal sectors means one chance in four, with nothing to measure. An experimental probability comes from what actually happened in a limited run of trials, and it wobbles — Q9's spinner lands on red rather more often than an even split would suggest, and nothing at all is wrong with the spinner.

So the last part asks which value to quote. When the structure is known and the outcomes really are equally likely, quote the theoretical one; the experiment is then evidence that supports it, not a competitor. The experimental value is what you fall back on when the structure is unknown or the outcomes are not equally likely. (This question carries its results in a table, so it lives on the printed sheet rather than on this page.)

Naming the type of an event (Q10)

Certain when it contains every outcome, so its probability is 1. Impossible when it contains none, so its probability is 0. Elementary when it contains exactly one. Probable when it may or may not happen. The instruction in Q10 is to give the most precise name that applies, which is the whole difficulty: a single-outcome event is both probable and elementary, and only one of those two answers says something.

Its last part asks whether "even" and "odd" are complementary, and a complete answer checks both conditions — no outcome is in both, and together they cover the sample space — rather than just observing that the probabilities happen to add to 1.

Q11: the synthesis question

A vending machine choosing a flavour and a size independently, which is a two-step experiment dressed as one action. Write the sample space, then answer four questions of four different kinds on it: a union with an overlap to subtract, a compatibility judgement, and a complementary event to name and evaluate. Every one of them is answered by pointing at outcomes in the list you wrote first — which is the argument for writing it out properly.

Getting the most out of it

List the sample space before anything else

Every question on this sheet is answered from the list. Write it with a tree, a table or set braces, use pairs in a fixed order, and count how many outcomes it holds. Questions that look hard are almost always questions where the list was skipped.

Ask whether the outcomes are equally likely

Before dividing, check that no outcome in your list is more likely than another. If they are not — and totals, sums and categories are the usual offenders — go back to the underlying outcomes and build the list from those instead. This single check is what separates a probability from a guess.

Say the words, not just the fraction

Describe the experiment, the sample space and the event in sentences; name the type of an event precisely; explain why a count doubles or why a union is not a sum. These questions are written to be answered in prose with numbers in it, and the reasoning is the part being assessed.

Common questions