Question

an airline gathers data on late departures and early arrivals over a month. it finds that the probability of a late departure is 12 percent, the probability of an early arrival is 27 percent, and the probability of both a late departure and an early arrival is 4 percent. which equation shows how to correctly calculate the probability of a late departure or an early arrival?

• ( p(\text{late departure or early arrival}) = 0.12 + 0.04 )
• ( p(\text{late departure or early arrival}) = 0.12 + 0.27 )
• ( p(\text{late departure or early arrival}) = 0.12 + 0.27 - 0.04 )
• ( p(\text{late departure or early arrival}) = 0.27 + 0.04 - 0.12 )

Explanation:

Step1: Recall the principle of inclusion - exclusion for probabilities

The formula for the probability of the union of two events \( A \) and \( B \) is \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \). Let \( A \) be the event of a late departure and \( B \) be the event of an early arrival.

Step2: Identify the values of \( P(A) \), \( P(B) \) and \( P(A\cap B) \)

We know that \( P(A) = 0.12 \) (probability of late departure), \( P(B)=0.27 \) (probability of early arrival) and \( P(A\cap B) = 0.04 \) (probability of both late departure and early arrival).

Step3: Substitute the values into the formula

Substituting these values into the formula \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \), we get \( P(\text{late departure or early arrival})=0.12 + 0.27-0.04 \).

Answer:

\( P(\text{late departure or early arrival}) = 0.12 + 0.27 - 0.04 \) (the third option)