Secondary 3 Probability Worksheet

All 4 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Geometric Probability

   In each situation a point is chosen at random inside the object described. Find the probability asked for, and say whether you compared lengths, areas or volumes.

   1. A guardrail along a boardwalk in Rimouski is $20\text{ m}$ long. The stretch between the $8\text{ m}$ mark and the $13\text{ m}$ mark has been repainted. A gull lands at a random point of the guardrail. What is the probability that it lands on the repainted stretch?
   2. A cork board in a school hallway measures $120\text{ cm}$ by $80\text{ cm}$. A poster measuring $40\text{ cm}$ by $30\text{ cm}$ is pinned flat on it. A thumbtack is pushed in at a random point of the board. What is the probability that it goes through the poster? Give the answer as a fraction and as a percentage.
   3. A rectangular tank has a base measuring $60\text{ cm}$ by $40\text{ cm}$, and it stands $30\text{ cm}$ deep. It is completely full of water, and a tiny air bubble forms at a random point of the water. What is the probability that it forms in the bottom $10\text{ cm}$ of water?
2. Q2Permutations, Arrangements and Combinations

   A bubble-tea shop in Trois-Rivières has five flavours: mango (A), taro (B), matcha (C), lychee (D) and melon (E).

   1. The owner will put three of the five flavours on a poster, one under the other, so a first, a second and a third are named. Reason out how many different posters are possible. Explain your reasoning; do not use a formula.
   2. Instead, she chooses three of the five flavours to put on special. This time the order means nothing — “A, B, C” and “C, A, B” are the same special. How many different specials are possible? Explain how this answer is related to the one in a).
   3. List all the specials from b), using the letters, to check your count.
3. Q3Probability

   Three situations are described below.

   1. One card is drawn from a shuffled set of $12$ cards numbered $1$ to $12$. Find the probability that the number drawn is a multiple of $4$.
   2. A raindrop falls on a rectangular tarp measuring $2\text{ m}$ by $5\text{ m}$, landing at a random point of it. A square patch measuring $1\text{ m}$ by $1\text{ m}$ is painted in the middle of the tarp. Find the probability that the drop lands on the patch.
   3. A bottle cap is flipped $400$ times and lands upright $148$ times. Find the probability that it lands upright.

   For each one, name the kind of probability you used — counting the equally likely outcomes of a sample space, comparing measurements of a figure, or using the results of repeated trials. Then say which of the three probabilities could come out differently if the whole situation were carried out again tomorrow, and why.

4. Q4Synthesis — drawing on several sheets in this topic

   A mural on a school wall is a rectangle $6\text{ m}$ wide and $4\text{ m}$ high, ruled into a grid of squares measuring $1\text{ m}$ by $1\text{ m}$.

   1. How many squares does the grid contain?
   2. The four squares forming a $2\times 2$ block in one corner are painted gold. A drop of paint falls at a random point of the mural. Find the probability that it lands on gold.
   3. The art teacher will sign two of the six squares along the bottom row. The order of the two signatures does not matter. Reason out how many different pairs of squares she could sign, then list them, numbering the bottom-row squares $1$ to $6$ from the left.
   4. She picks one of those pairs at random. Find the probability that the two signed squares are side by side.

The 4 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.

Three kinds of probability, and Q3 asks you to name which is which

The sheet does not treat probability as one technique. Q3 puts three situations side by side — a card drawn from a numbered set, a raindrop landing on a tarp, a bottle cap flipped four hundred times — and asks you to name the kind of probability each one needs, then to say which of the three answers could come out differently if the whole thing were done again tomorrow.

Geometric probability: decide first what you are comparing

Q1 runs the same idea at three different dimensions on purpose — a repainted stretch of a guardrail, a poster pinned to a cork board, a layer of water in a tank — and asks each time whether you compared lengths, areas or volumes. The arithmetic is a single division; naming the comparison is the skill.

Counting: does the order matter, and can something repeat?

Q2 is the whole idea in one question. Five flavours; three of them go on a poster, one under the other, so a first, a second and a third are named — then the same five, with three chosen to go on special, where "A, B, C" and "C, A, B" are the same thing. The two answers are different, the question asks you how they are related, and it asks you to reason it out and not use a formula.

The challenge question adds a constraint: five friends in a row of cinema seats, two of whom want to sit together. The productive strategy is to glue the pair into a single block, arrange the four objects that leaves, then multiply by the two ways the pair can sit within the block — and the question asks you to check your count a second way, because a counting argument you cannot verify is a counting argument you cannot trust. From there the probability is that count over the total number of orders.

Equally likely is an assumption, not a default

The spinner question is the one to read even if you skip everything else. Three colours, so a student announces that each has probability one third — and the sectors are 180, 120 and 60 degrees. Counting outcomes is only valid when the outcomes are equally likely, and here they are plainly not; this is a geometric probability wearing the costume of a counting one, and each colour's probability is its share of the full turn.

The synthesis question

A mural ruled into a grid of square metres, with a block of squares painted gold. It starts as a counting problem, becomes a geometric one when a drop of paint lands at a random point, and then turns into a counting problem again: how many pairs of squares along the bottom row a teacher could sign, when the order of the two signatures does not matter. You are asked to reason the count out and then list the pairs, and the last part uses that list — of all the possible pairs, how many put the two signed squares side by side.

That final step is the shape most probability questions take from here on. Count the whole set of possibilities, count the ones with the property you care about, and divide. The counting is the hard part; the probability is the division at the end.

Getting the most out of it

Ask two questions before you calculate anything

Are the outcomes equally likely, and does the order matter? The first decides whether you may count at all or must compare measurements instead; the second decides whether two answers differ by a factor you have to divide out. Nearly every error on this sheet is one of those two questions answered without being asked.

Draw the figure and shade the favourable region

For the geometric questions, a sketch with the measurements written on it makes the comparison obvious and makes a unit mismatch visible. Write both measurements down as a fraction before you simplify anything, so you can see that they are the same kind of quantity.

List them when you can, count them when you cannot

Where a question asks for a list, do the list — it is the only way to check a count. Where the numbers are too big to list, count a smaller version of the same problem by hand first and confirm your reasoning reproduces it. That habit is what makes counting arguments trustworthy without a formula to fall back on.

Sanity-check every probability

It has to sit between 0 and 1, and it should feel right: a small patch on a big board should give a small number. A probability greater than 1 usually means the two measurements were the wrong way up, or in different units.

Common questions