A teacher is selected at random - AQA - A-Level Maths Mechanics - Question 13 - 2020 - Paper 3

To find the probability that the teacher is not a sixth-form teacher, we need to first find the number of teachers who are not in the sixth-form category.

The total number of sixth-form teachers is:

• Male: 23
• Female: 24

Hence, total sixth-form teachers = 23 + 24 = 47.

Therefore, the number of teachers not in the sixth-form = 200 - 47 = 153.

Now, the probability is:

$P(\text{not sixth-form}) = \frac{\text{Number of not sixth-form teachers}}{\text{Total number of teachers}} = \frac{153}{200}$

We start by finding the total number of male teachers:

• Primary: 9
• Secondary: 24
• Sixth-form: 23

Total male teachers = 9 + 24 + 23 = 56.

To find those who are not in primary, we subtract the number of primary male teachers:

Total male not primary = 24 + 23 = 47.

Hence, the conditional probability is:

$P(\text{not primary | male}) = \frac{\text{Number of male not primary}}{\text{Total male teachers}} = \frac{47}{56}$

To find the probability that all three randomly chosen teachers are secondary teachers, we first determine the total number of secondary teachers:

• Male: 24
• Female: 85

Total secondary teachers = 24 + 85 = 109.

Now, the probability that all three chosen teachers are secondary is given by:

$P(\text{all secondary}) = \frac{109}{200} \times \frac{108}{199} \times \frac{107}{198}.$

Calculating this gives:

$P(\text{all secondary}) \approx 0.16.$

Thus, the final answer is approximately 0.16.