To find the probability of either a late departure or an early arrival, we must consider the probabilities of both events, as well as the intersection where both events happen simultaneously.
We are given:
- The probability of a late departure ([tex]\(P(\text{Late Departure})\)[/tex]) is 12 percent, or 0.12.
- The probability of an early arrival ([tex]\(P(\text{Early Arrival})\)[/tex]) is 27 percent, or 0.27.
- The probability of both a late departure and an early arrival ([tex]\(P(\text{Late Departure} \cap \text{Early Arrival})\)[/tex]) is 4 percent, or 0.04.
The correct formula to calculate the probability of either a late departure or an early arrival involves the principle of inclusion and exclusion:
[tex]\[ P(\text{Late Departure} \cup \text{Early Arrival}) = P(\text{Late Departure}) + P(\text{Early Arrival}) - P(\text{Late Departure} \cap \text{Early Arrival}) \][/tex]
Putting the given probabilities into this equation:
[tex]\[ P(\text{Late Departure} \cup \text{Early Arrival}) = 0.12 + 0.27 - 0.04 \][/tex]
Carrying out the arithmetic:
[tex]\[ P(\text{Late Departure} \cup \text{Early Arrival}) = 0.35000000000000003 \][/tex]
Therefore, the correct choice, which shows how to accurately calculate the probability of a late departure or an early arrival, is:
[tex]\[ P(\text{Late Departure or Early Arrival}) = 0.12 + 0.27 - 0.04 \][/tex]
This corresponds to the third option from the given list.
