Mathematics 9709/62 — February/March 2019

Walkthrough

There are two mutually exclusive ways for Tamar to wear red socks: blue suit with red socks, or grey suit with red socks. Since these cannot both happen on the same day, their probabilities are added. Within each way, the suit choice and the socks choice must both happen, so multiply the probabilities along the branch. The blue branch gives $0.6 \times 0.2 = 0.12$; the grey branch gives $0.4 \times 0.32 = 0.128$. Adding these gives $0.248$. This is the law of total probability applied to the two possible suit colours.

Key Takeaways

This question tests the addition rule for mutually exclusive outcomes and the multiplication rule for probability of two events both happening. It is equivalent to using a tree diagram: label branches with $B$ and $G$, then each branch has a red-socks sub-branch with the given conditional probability. The total probability of red socks is the sum of the two compound-branch probabilities.

Common Mistakes

• Adding $0.2$ and $0.32$ directly, ignoring the probabilities of the suits.
• Using the suit probabilities incorrectly, e.g. $0.6 + 0.4$, or multiplying all four numbers together.
• Forgetting that the two events (suit colour and sock colour) are combined by multiplication because they must both occur.

Things to Be Careful About

• The conditional probabilities are given as $P(R \mid B)$ and $P(R \mid G)$, so they must each be multiplied by the corresponding suit probability.
• The answer can be written as $0.248$ or $\frac{31}{125}$; keep the exact fraction if you will use it again in part (ii).
• Do not round the intermediate value before using it in part (ii).

Walkthrough

We are conditioning on the event 'not wearing red socks', so use the formula

For the numerator, the probability of a grey suit and no red socks is the probability of grey suit times the probability of no red socks given a grey suit: $0.4 \times (1 - 0.32) = 0.4 \times 0.68 = 0.272$. For the denominator, the total probability of no red socks is the complement of the part (i) answer: $1 - 0.248 = 0.752$. Equivalently, it is $0.6 \times 0.8 + 0.4 \times 0.68$, because blue suits come with no red socks with probability $0.8$, and grey suits come with no red socks with probability $0.68$. Dividing gives $0.272 / 0.752$, which simplifies to $17/47 = 0.362$.

Key Takeaways

This part tests conditional probability together with the complement rule. It also shows that before conditioning, one must find the total probability of the conditioning event. The denominator is not just the probability of a suit; it must be the total probability of 'not red socks'.

Common Mistakes

• Using $P(G) = 0.4$ as the numerator when the condition on not wearing red socks is ignored.
• Using $P(R) = 0.248$ rather than $P(R') = 1 - 0.248$ in the denominator.
• Writing $0.4 \times 0.32$ for the numerator; that is the probability of grey suit and red socks, not grey suit and no red socks.
• Quoting the final decimal after rounding without showing the fraction setup, which may not earn method marks.

Things to Be Careful About

• Show the fraction with the unsimplified numerator and denominator, e.g.

so the method is clear and matches the mark scheme.

• If using their part (i) result, use the exact value (or enough decimal places) so the final answer rounds correctly.
• Remember the condition changes the sample space: the denominator is smaller than 1, and only the 'not red socks' days are considered.