11.1 The letters of the word EQUATION are randomly used to form a new word consisting of five letters - NSC Mathematics - Question 11 - 2017 - Paper 1

Given that events A and B are independent, we can calculate P(A or B) using the formula:

$P(A or B) = P(A) + P(B) - P(A and B)$

Since P(A) and P(B) are independent, we have:

$P(A and B) = P(A) \times P(B) = \frac{2}{5} \times 0.35 = 0.14$

Now substituting the values:

$P(A or B) = \frac{2}{5} + 0.35 - 0.14$

Calculating further:

$P(A or B) = 0.4 + 0.35 - 0.14 = 0.61$

Therefore, P(A or B) = 0.61.

To find the probability that a learner did not attend School A, we consider the proportions:

Since 20% attend School A, the probability that a learner did not attend School A is:

$P(not A) = 1 - P(A) = 1 - 0.2 = 0.8$

Thus, the probability is 0.8.

From the data, we see that the percentage of learners attending School B is 30% with a pass rate of 50%. Thus, the failure rate is:

$0.5\text{ (failure rate)} = 1 - 0.5 = 0.5$

Now, the joint probability is:

$P(B and fail) = P(B) \times P(fail | B) = 0.3 \times 0.5 = 0.15$

Hence, the probability is 0.15.

To find the probability that a learner passed Grade 12, we will consider the pass rates from all three schools, weighted by their respective attendance:

Using the law of total probability:

$P(pass) = P(A) \times P(pass | A) + P(B) \times P(pass | B) + P(C) \times P(pass | C)$

Substituting the respective values:

$P(pass) = 0.2 \times 0.8 + 0.3 \times 0.5 + 0.5 \times 0.9$

Calculating:

$P(pass) = 0.16 + 0.15 + 0.45 = 0.76$

Thus, the probability that a learner passed Grade 12 in 2016 is 0.76.