McGill University · MATH 123

McGill MATH 123: Linear Algebra and Probability tutoring

Linear Algebra and Probability: one-to-one online tutoring and exam preparation.

Online 60 min: $55 CAD · Pricing Midterm & final exam help Try a free practice question Free lesson: conditional probability and Bayes

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Course overview

McGill MATH 123 combines vector and matrix methods, linear programming and introductory probability for Management students.

Restricted to Desautels Faculty of Management students.

MATH 123 topics to review

Vectors, lines, planes and dot/cross productsLinear systems, matrix operations and rankLinear dependence and independence, inverses and determinantsLinear programming with tableauxSample spaces, events and probabilityConditional probability and Bayes’ ruleRandom sampling, random variables and common distributions

Common difficulties

Connecting matrix calculations to the original system
Translating an optimization problem into an objective and constraints
Keeping the conditioning event clear in a probability calculation

Preparing for this course

Review linear equations, inequalities, fractions and the meaning of an event before working through the examples below. The short check tests conditional sampling; the worked example tests how to build and verify a linear program. These are two focused skills, not a measure of readiness for the whole course. Use your current section’s outline to choose the rest of your practice.

Registration requirements

McGill lists MATH 123 as open to Faculty of Management students only. It is not open to students who have taken or are taking MATH 133, CEGEP objective 00UQ or an equivalent. The listing directs Management students planning a major or minor offered by the Department of Mathematics and Statistics to MATH 133 instead. Confirm current eligibility and registration details with McGill.

Check the official course catalogue

Quick check: a second draw without replacement

A box contains 10 distinct items: 4 defective and 6 working. Draw two items one after the other, without putting the first back. At each draw, every remaining item has the same chance of being chosen. You learn that the first item was defective. Let $D_1$ mean “the first item is defective” and $D_2$ mean “the second item is defective.” Find $P(D_2\mid D_1)$ , the probability that the second item is defective given that the first was defective. Explain why $4/10$ is not the conditional probability.

Show a hint

Apply the information after the vertical bar first. Remove one defective item, then count both the defective items and all the items still in the box.

Show the reasoning

Update the count for any first defective item
Whichever defective item was drawn first, 9 items remain: 3 defective and 6 working. Because every remaining item is equally likely to be chosen, there is a $3/9$ chance of a defective second item. The same fraction applies whichever first defective item was removed.
Compare conditional and unconditional probabilities
Therefore $P(D_2\mid D_1)=3/9=1/3$ . Before learning the first result, each of the 10 items is equally likely to be second, so $P(D_2)=4/10$ . Since $P(D_1)=4/10>0$ , independence would require $P(D_2\mid D_1)=P(D_2)$ . Here $1/3\ne4/10$ , so the two events are not independent. The fraction $4/10$ ignores the information about the first draw. These probabilities have no units.

Worked example: meet two requirements at minimum cost

A hypothetical manufacturer can buy two bulk blends, A and B. Each kilogram of A supplies 2 g of component R and 1 g of component S; each kilogram of B supplies 1 g of R and 3 g of S. A costs 3 CAD/kg and B costs 4 CAD/kg. The purchase must supply at least 8 g of R and at least 9 g of S. Find the least-cost purchase. Assume quantities are divisible, prices and composition are constant, excess components are allowed, and there are no other limits or charges.

1. Define quantities, constraints and cost

Let $x$ and $y$ be the numbers of kilograms of A and B purchased, respectively, where $x,y\ge0$ are real numbers. The numerical component totals in grams are $2x+y$ for R and $x+3y$ for S. “At least” gives $2x+y\ge8$ and $x+3y\ge9$ . Minimize the numerical value of the cost $C=3x+4y$ in CAD. The coefficients in the component formulas are in g/kg; the price coefficients are in CAD/kg.

2. Find a candidate where both requirements are exact

Solve $2x+y=8$ and $x+3y=9$ . The first equation gives $y=8-2x$ . Substitute into the second: $x+3(8-2x)=9$ , so $-5x=-15$ . Thus $x=3$ and $y=2$ . Meeting both boundaries gives a candidate, but does not yet prove minimum cost.

3. Check the candidate in the original model

The masses 3 kg of A and 2 kg of B are nonnegative. They supply $2(3)+2=8$ g of R and $3+3(2)=9$ g of S, meeting both minimum requirements. Their cost is $3(3)+4(2)=17$ CAD.

4. Prove that no feasible purchase costs less

Using the numerical quantities in the stated units, rewrite the cost as $C=(2x+y)+(x+3y)$ . Every purchase that meets all the requirements therefore satisfies $C\ge8+9=17$ . Our candidate attains that lower bound, so 17 CAD is the global minimum. Equality requires both component totals to equal their minimums; the two equations have the unique solution $(3,2)$ , so this is the unique least-cost purchase.

Try it yourself

Keep the same model. Someone proposes buying only 4 kg of A because it supplies the required 8 g of R. Is this purchase feasible? Check both components before comparing costs.

Show the explained solution

Here $x=4$ and $y=0$ . The purchase supplies $2(4)+0=8$ g of R, but only $4+3(0)=4$ g of S, below the required 9 g. Its 12 CAD cost cannot be compared as a feasible alternative: it fails the S constraint.

This example develops a two-variable model and a direct optimality check. It does not teach the tableau method. Use your section’s materials for the required linear-programming algorithm, and bring any unclear constraint or tableau step to a tutoring session.

How tutoring can help

You can start with a question alone. Your tutor can help you set up a matrix and explain how its row operations relate to the original equations, translate a linear-programming constraint while checking its units and direction, or identify the outcomes left after a probability condition. You can then try a fresh problem with guidance and explain the setup and verification yourself.

Exam preparation for MATH 123

Use your section’s current outline and assigned practice to balance matrices, vector geometry, linear programming and probability. In each problem, state the model or events before calculating, then check the result against the original conditions. These examples do not tell you what each assessment will cover, its timing or which methods are allowed. Follow your instructor’s current instructions.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

Independent practice

Free practice and study guides for MATH 123

18 min

Conditional Probability and Bayes: Choose the Right Denominator

Start with counts, distinguish “given” from “and,” derive Bayes’ formula and test independence with a frequency tree and four explained exercises.

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