Mathematics 9709/51 — October/November 2022

Walkthrough

We are given a probability distribution table for a discrete random variable $X$, with two unknown probabilities $p$ and $q$. There are two standard facts we can use to find them.

First, the probabilities in any probability distribution must add up to $1$. This gives the equation

which simplifies to $p + q = 0.42$.

Second, the expectation is defined as

We substitute each value of $x$ and its probability, using $p$ and $q$ for the unknown probabilities. This gives

After simplifying the constant terms, we obtain

Now we have two linear equations in $p$ and $q$. We solve them simultaneously. From $p + q = 0.42$, we can write $q = 0.42 - p$ and substitute into the second equation. This gives a single equation in $p$:

which simplifies to $-1.5p + 0.21 = -0.24$, so $p = 0.3$. Substituting back gives $q = 0.12$.

Key Takeaways

• For any discrete probability distribution, the sum of all probabilities is always $1$.
• The expectation formula $E(X) = \sum x \, P(X = x)$ is the key tool for using a given mean to find unknown probabilities.
• Two unknown probabilities require two independent equations: one from the total probability and one from the expectation.
• Solving simultaneous linear equations is a core algebraic skill needed in probability and statistics.

Common Mistakes

• Forgetting to include all probabilities in the sum-to-1 equation, or incorrectly adding the constants.
• Making sign errors when computing $E(X)$, especially with negative values of $x$.
• Using the wrong signs when simplifying $-p + 0.5q = -0.24$.
• Not simplifying the expectation equation before attempting to solve, leading to arithmetic errors.
• Stopping after finding only one of the two unknowns.

Things to Be Careful About

• The mark scheme awards marks for forming both equations: the sum-of-probabilities equation and the expectation equation. You must show both clearly.
• The expectation equation may be left unsimplified and still earn the method mark, but simplifying helps avoid mistakes.
• When solving, be careful with decimal arithmetic. Multiplying the equations by 10 or 2 can make the algebra easier.
• Always check that the final values are sensible: probabilities must lie between $0$ and $1$, and they should satisfy both original equations.