**Question 7**:

For this question, give all your answers as fractions..

When Ivan goes to shopping mall, the probability that he wears a hat is 5/8. If he wears a hat, then the probability that he wears sport shoes is 2/3. If he does not wear a hat, then the probability that he wears sport shoes is 1/6.

**(a)**Diagram 4 in the answer space shows all the possible outcomes. Complete the tree diagram in Diagram 4. [2 marks]

**(b)**Hence, find the probability that Ivan wears a hat or sport shoes but not both. [2 marks]

**Solution**:

**(a)**

**(b)**

$\begin{array}{l}\text{Probability that Ivanwears a hat or sport shoes}\\ =P\left(\text{Hat,no sport shoes}\right)+P\left(\text{No hat,wears sport shoes}\right)\\ =\left(\frac{5}{8}\times \frac{1}{3}\right)+\left(\frac{3}{8}\times \frac{1}{6}\right)\\ =\frac{13}{48}\end{array}$

**Question 8**:

Firdaus is training for a marathon. He practises according to his routine which is running as far as 32 km a week. He plans to increase his running distance,

*D*(

*x*), in km, each week by 10% of his running distance in the previous week, where

*x*represents number of weeks of training.

Derive a mathematical model for his running distance,

*D*(

*x*) through mathematical modeling.

[4 marks]

**Solution**:

$10\%=\frac{10}{100}=0.1$

*D*(

*x*) = 32(1.1)

^{x}

Week | Calculation (km) | Running distance (km) |

1 | 32 + 32 × 0.1 = 32(1 + 0.1) | 3.2 (1.1) |

2 | 32(1.1) + 32(1.1) × 0.1 = 32(1.1)(1 + 0.1) | 3.2 (1.1)^{2} |

3 | 32(1.1)^{2} + 32(1.1)^{2} × 0.1 = 32(1.1)^{2} (1 + 0.1) | 3.2 (1.1)^{3} |

4 | 32(1.1)^{3} + 32(1.1)^{3} × 0.1 = 32(1.1)^{3} (1 + 0.1) | 3.2 (1.1)^{4} |