Mathematics 9709/52 — October/November 2022

Walkthrough

Kino can travel by bus, by car or on foot, and these are the only possible ways. The probability that he is late on any day is the weighted sum of the probabilities of being late for each travel method. For the bus, this is $0.2 \times x$; for the car, $0.1 \times 2x$; for walking, $0.7 \times 0.25$. Add these and set the sum equal to $0.235$. This gives $0.4x + 0.175 = 0.235$, so $0.4x = 0.06$ and $x = 0.15$.

Key Takeaways

This question tests the law of total probability: when events are mutually exclusive and exhaustive, the total probability is the sum of the products of the probability of each event and the conditional probability of the outcome given that event. It also tests solving a simple linear equation.

Common Mistakes

• Forgetting that the car late probability is $2x$, not $x$.
• Omitting the walking term $0.7 \times 0.25$.
• Setting the expression equal to $1$ instead of $0.235$.
• Mis-collecting the $x$-terms, for example writing $0.2x + 0.1x$ instead of $0.2x + 0.2x$.

Things to Be Careful About

The travel probabilities sum to $1$, so the events are exhaustive. The late probabilities are conditional probabilities, not travel probabilities. Keep the values in decimal form; then $0.4x = 0.06$ gives $x = 0.15$.

Walkthrough

We need $P(\text{car} \mid \text{not late})$. By the conditional probability formula, this is:

From part (a), $x = 0.15$, so the probability of being late by car is $2x = 0.30$. Therefore the probability of going by car and not being late is $0.1 \times (1 - 0.30) = 0.07$. The probability of not being late is $1 - 0.235 = 0.765$. Hence the conditional probability is:

to 3 significant figures.

Key Takeaways

This question tests conditional probability and the complement rule. The joint probability of two events is found by multiplication, and the denominator of a conditional probability must be the probability of the condition being given.

Common Mistakes

• Using $0.1 \times 0.3$ instead of $0.1 \times (1 - 0.30)$.
• Using $0.235$ as the denominator instead of $1 - 0.235$.
• Using $x$ instead of $2x$ for the car late probability.
• Leaving the answer as an unsimplified fraction such as $0.07/0.765$ when the question expects a decimal to at least 3 significant figures.

Things to Be Careful About

The denominator is the probability of being not late, so it is $1 - 0.235$. The numerator must be the joint probability of both car and not late. If $x$ was found incorrectly, the mark scheme allows follow-through with the candidate's value, but the correct value is $x = 0.15$.