Fifteen boys go hiking. Five get lost, eight get sunburned, and six return home without problems. (a) What is the probability that a sunburned boy got lost? (b) What is the probability that a lost boy got sunburned?

Conditional probability is the likelihood of an event occurring given that another event has already happened.
It's written as \( P(A|B) \) and reads 'the probability of A given B'.
To compute conditional probability, we use the formula:
\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \].
For instance, in the exercise, we are asked to find the probability that a sunburned boy also got lost. Here, 'the boy got sunburned' is event B, and 'the boy got lost' is event A.
We already know that 4 boys were both sunburned and lost, which is our \( P(A \text{ and } B) \). Dividing this by the total number of sunburned boys gives our solution: \( P(\text{Lost | Sunburned}) = \frac{4}{8} = \frac{1}{2} \).
Similarly, to find the probability that a lost boy also got sunburned, event B becomes 'the boy got lost,' and event A is 'the boy got sunburned':
\( P(\text{Sunburned | Lost}) = \frac{4}{5} \).

The inclusion-exclusion principle is used to calculate the number of items in overlapping sets.
It helps to avoid double-counting those items that are in more than one set.
The principle states that:
\[ |A \text{ or } B| = |A| + |B| - |A \text{ and } B| \] .
For instance, when summing our lost and sunburned boys, we get 5 + 8 = 13. However, we know the total number of boys is only 15, and 6 boys faced no issues.
Using the inclusion-exclusion principle, we calculate the number of boys who got both sunburned and lost:
\[ 5 + 8 - (15 - 6) = 13 - 9 = 4 \] .
Ensuring calculations like these help in accounting for the fact that some boys fall into overlapping categories so that accurate probabilities can be determined.

Event overlap occurs when two or more events share common outcomes.
In our exercise, 'getting lost' and 'getting sunburned' are two events, and some boys experienced both.
Overlapping events were identified when we used the inclusion-exclusion principle:
5 boys lost + 8 boys sunburned - (15 total boys - 6 with no issues) = 4 boys both lost and sunburned.
Overlapping events complicate probability calculations because simply summing individual probabilities would result in overcounting.
Instead, we adjust for these by subtracting the overlap from the total during our inclusion-exclusion calculations.