A school has 700 learners. 72% use cricket nets, 48% use tennis courts, and 8% use neither. Find the number of learners using at least one facility and the probability a learner uses exactly one facil

Total number of learners = 700.
Let C be the event that a learner uses cricket nets.
Let T be the event that a learner uses tennis courts.
$P(C) = 72\% = 0.72$
$P(T) = 48\% = 0.48$
$P(\text{neither}) = P(C' \cap T') = 8\% = 0.08$

From $P(C' \cap T') = 0.08$, we know that the probability of using at least one facility is:

We also use the formula for the union of two events:

This is the probability of using both facilities.

Determine the number of learners who use at least one of the facilities.
The probability of using at least one facility is $P(C \cup T) = 0.92$.
Number of learners = $P(C \cup T) \times \text{Total learners}$
Number of learners = $0.92 \times 700$
Number of learners = 644
The number of learners who use at least one facility is $\boxed{\text{644}}$.

Determine the probability that a learner uses exactly one facility.
The probability of using exactly one facility is $P(C \text{ only}) + P(T \text{ only})$.
$P(C \text{ only}) = P(C) - P(C \cap T) = 0.72 - 0.28 = 0.44$
$P(T \text{ only}) = P(T) - P(C \cap T) = 0.48 - 0.28 = 0.20$
$P(\text{exactly one facility}) = 0.44 + 0.20 = 0.64$
Alternatively, $P(\text{exactly one facility}) = P(C \cup T) - P(C \cap T) = 0.92 - 0.28 = 0.64$.
The probability that a learner uses exactly one facility is $\boxed{\text{0.64}}$.

Determine whether the events 'using the cricket nets' and 'using the tennis courts' are independent or not.
Two events A and B are independent if $P(A \cap B) = P(A) \times P(B)$.
Here, we check if $P(C \cap T) = P(C) \times P(T)$.
We found $P(C \cap T) = 0.28$.
Now, calculate $P(C) \times P(T)$:

Since $0.28 \neq 0.3456$, the events are not independent.
The events are $\boxed{\text{not independent}}$.

The table shows data from the NSFAS financial aid office.
Total students = 10 730.

Determine the probability that a student selected at random is a student receiving financial aid.
This is $P(A)$.

The probability that a student is receiving financial aid is $\boxed{\frac{6101}{10730} \text{ or approximately } 0.5686}$.

Determine the probability that a student selected at random is a repeating learner and is not receiving financial aid.
This is $P(R \cap A')$.
From the table, the number of repeating recipients not receiving financial aid is 731.

The probability that a student is a repeating learner and is not receiving financial aid is $\boxed{\frac{731}{10730} \text{ or approximately } 0.0681}$.

Determine the probability that a student selected at random is a first-time learner and is receiving financial aid.
This is $P(F \cap A)$.
From the table, the number of first-time recipients receiving financial aid is 4222.

The probability that a student is a first-time learner and is receiving financial aid is $\boxed{\frac{4222}{10730} \text{ or approximately } 0.3935}$.