Secondary 4 Mathematical Expectation and Odds Worksheet

6 of the 7 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 6 of the 7 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. Q1Conditional Probability

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

2. Q2Geometric Probability –

   A rectangular skate park measures $30\text{ m}$ by $18\text{ m}$. Inside it, a circular bowl of radius $6\text{ m}$ is cut into the concrete. A leaf blown by the wind lands at a point chosen at random in the park, every point being equally likely. Find the probability that the leaf lands in the bowl, to the nearest thousandth.

3. Q3Mathematical Expectation

   At a maple festival booth, a player pays $3 to spin a wheel once. The wheel pays $10 with probability $0.1$, pays $4 with probability $0.3$, and pays nothing with probability $0.6$. Find the mathematical expectation of a player's net gain for one spin, and say whether the game is fair.

4. Q4The Odds For and the Odds Against

   On a busy route, the odds that a ferry leaves on time are $9:5$ for.

   1. What is the probability that the ferry leaves on time?
   2. State the odds against the ferry leaving on time.
5. Q5The Types of Probability –

   For each statement, say whether the probability given is theoretical, experimental (frequency-based) or subjective, and justify your choice in one line.

   1. “This 12-sided die is perfectly balanced, so the probability of rolling a multiple of 4 is $\frac{1}{4}$.”
   2. “Of the 500 loaves we baked last month, 35 came out underbaked, so the probability that the next loaf is underbaked is $0.07$.”
   3. “I think our team has a $70\%$ chance of winning Saturday — the players looked confident at practice.”
6. Q6Types of Events –

   A box contains 20 tokens numbered from 1 to 20, and one token is drawn at random. Consider the events $A$: “the number is even”, $B$: “the number is a multiple of 5”, $C$: “the number is greater than 20”.

   1. Are $A$ and $B$ compatible or incompatible? Justify with the outcomes.
   2. What type of event is $C$, and what is its probability?
   3. Describe the complement of $A$ in words and give $P(A^{\prime })$.
7. Q7Synthesis — drawing on several sheets in this topic

   At a community fair, a beanbag is tossed onto a square target board measuring $1\text{ m}$ by $1\text{ m}$; it lands at a point chosen at random on the board. A circular zone centred on the board pays the player $6; anywhere else pays nothing. Each throw costs $2.

   1. The circular zone has radius $0.3\text{ m}$. Find the organizer's expected profit per throw, to the nearest cent.
   2. The organizer now wants the odds against a player winning the $6 to be exactly $3:1$. Find the radius of the zone he must use, to the nearest centimetre, and state his expected profit per throw when the odds are exactly $3:1$.

The 7 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.

Every probability question is a fraction. Get the denominator right first.

Almost everything on this sheet comes down to deciding what the denominator is, and each section of the unit changes it in a different way: a condition shrinks it, a geometric setting turns it into an area, odds replace it with something that is not a total at all. Fix the denominator and the rest is arithmetic.

Conditional probability: the condition shrinks the sample space (Q1)

P(A | B) is read "the probability of A given B", and the word "given" means B has already happened. Everything that is not B is no longer in play, so B becomes the new universe:

With a two-way table you rarely need the formula. The condition names one row or one column; that line's total becomes the denominator, and the single cell where the two categories meet becomes the numerator. Q1 conditions on holding a certificate, so the certificate column total is the denominator, not the 240 people surveyed. (This question depends on its table, so it lives on the printed sheet rather than on this page.)

Geometric probability: a ratio of measures (Q2, Q7)

When a point is chosen at random in a region and every point is equally likely, the probability of landing in a sub-region is the ratio of their sizes — lengths on a line, areas on a surface, volumes in a solid. There is no counting because there is nothing countable; the measure does the counting.

Two things to guard. Compare areas, not radii or diameters: a circle of radius 6 in a 30 by 18 rectangle covers 36π out of 540, and comparing 6 with 30 is meaningless. And make sure both measures are in the same unit before dividing — a probability is a pure number, so if the units do not cancel, something has gone wrong upstream.

Q7 runs the idea backwards, which is the harder direction and the one exams prefer: you are given the probability and asked for a dimension. Set up the same ratio, put the known probability on one side, and solve for the radius or the width. When that leaves a second-degree equation, expect one root to be geometrically impossible — a border wider than half the shorter side leaves no middle region at all — and reject it explicitly, with the reason written out.

Mathematical expectation: a long-run average, not a prediction (Q3, Q7)

Expectation is each outcome multiplied by its probability, all added up: E = Σ x · P(x). Set it out as two columns, outcomes and probabilities, and check the probabilities add to 1 before you multiply anything — a column that does not sum to 1 means a missing outcome, and the "nothing happens" outcome is the one people leave out.

The interpretation sentence matters as much as the number. An expectation is an average over many repetitions, so it is not a claim about any single trial. Nobody loses 80 cents on one spin — they lose the price of the spin or win a prize. What the expectation says is what happens per play across a long evening, which is why it is the right tool for comparing two options and the wrong tool for predicting one.

Odds are not probability (Q4, Q7)

This is the single most misread notation in the unit, and the fix is one sentence: probability compares favourable cases to the total, odds compare favourable cases to unfavourable ones.

Theoretical, experimental, subjective (Q5)

Three ways of arriving at a number, and the question is always about where the number came from, not whether it is right. Theoretical probability is counted from a model of equally likely outcomes, with no trials performed. Experimental probability is a relative frequency observed in actual trials. Subjective probability is a personal judgment — a hunch, an impression, an expert opinion — with no counting and no data behind it.

The trap is the model rather than the type. "Equally likely" is an assumption that has to be earned: five sectors on a spinner are equiprobable only if the five central angles are equal, and if they are not, the theoretical probabilities come from the angles. When experimental results disagree with a theoretical value, the honest first suspicion is that the model was wrong, not that the equipment is broken.

Types of events, and the word "independent" (Q6)

Two events are compatible when they can occur together — when their intersection is not empty — and incompatible when it is. An event with probability 0 is impossible, one with probability 1 is certain, and the complement A′ is "A does not happen", with P(A′) = 1 − P(A). Justify a compatibility claim by naming the outcomes in the intersection, as Q6(a) asks — listing them is the justification, and asserting the verdict alone is not.

Q7: the synthesis problem

A target board that chains three tools together. Part (a) is geometric probability feeding straight into an expectation: work out the chance of landing in the circle from the areas, use it to get the expected payout, then subtract the cost of a throw to reach the organiser's expected profit per throw. Part (b) reverses the chain — convert the required odds into a probability, put that probability into the area ratio, and solve for the radius before recomputing the profit. Check at the end that the circle you found actually fits on the board; a geometric answer that does not fit its own figure is not an answer.

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Getting the most out of it

Write the denominator before anything else

On every question, write down what the whole sample space is — all members, all of the board, the certificate holders only — before you write a numerator. Most wrong answers in this unit are right fractions over the wrong total.

Convert odds to a probability the moment you see them

Odds are a reporting format, not a working format. Turn every odds statement into a fraction as soon as you read it, do all the mathematics in probabilities, and convert back only if the question asked for odds.

Say the interpretation out loud

Several questions here ask for a verdict as well as a number: is the game fair, is the spinner defective, is one option better. Practise finishing every calculation with one sentence in plain words, because on an exam that sentence is a separate mark.

Common questions