Each morning, Ben goes to school by train. He takes either a local train or an express train. If he takes a local train, the probability that he is late is 0.15. If he takes an express train, the probability that he gets there on time is 0.98. The probability that he takes a local train is 0.40.

Two fair spinners are used in a game. Spinner $P$ has sectors labelled 1, 2 and 3. Spinner $Q$ has sectors labelled 2, 3 and 4. The score for each pair of outcomes is the product of the two numbers, as shown in Table 1.

Table 1

Scores are classified as starter, regular or premium.

• starter: 2, 3 or 4
• regular: 6 or 8
• premium: 9 or 12

These 9 outcomes are summarized in the cumulative frequency table, Table 2.

Table 2

Two fair spinners are used in a game. Spinner A has sectors labelled 1, 2 and 4. Spinner B has sectors labelled 2, 3 and 5. The total score for each pair of outcomes is shown in Table 1.

Table 1

Scores are classified as bronze, silver or gold.

• bronze: 3 or 4
• silver: 5 or 6
• gold: 7 or 9

These 9 outcomes are summarized in the cumulative frequency table, Table 2.

Table 2

On a school day, the probability that Maya catches the metro is $\frac{2}{3}$. If Maya catches the metro, the conditional probability that she arrives before registration is $r$. If Maya misses the metro, the conditional probability that she arrives before registration is $\frac{1}{4}$. The overall probability that Maya arrives before registration is $\frac{1}{2}$.