Question

Two different airlines have a flight from Los Angeles to New York that departs each weekday morning at a certain time. Let \(E\) denote the event that the first airline's flight is fully booked on a particular day, and let \(F\) denote the event that the second airline's flight is fully booked on that same day. Suppose that \(P(E)=.7, P(F)=.6\), and \(P(E \cap F)=.54\). a. Calculate \(P(E \mid F)\) the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. b. Calculate \(P(F \mid E)\).

Short answer

The probability that the first airline's flight is fully booked given that the second airline's flight is fully booked is 0.9. The probability that the second airline's flight is fully booked given that the first airline's flight is fully booked is 0.7714.

Step-by-step solution

Define Notation

Start by defining the probability of each event and the intersection of these events. Here, \(P(E)=0.7\) is the probability of the first airline's flight being fully booked, \(P(F)=0.6\) is the probability of the second airline's flight being fully booked and \(P(E ∩ F)=0.54\) is the probability of both flights being fully booked.

Calculate \(P(E \mid F)\)

To calculate the conditional probability, we need to apply the formula \(P(E \mid F) = P(E ∩ F) / P(F)\). Inserting given probabilities, we get \(P(E \mid F) = 0.54 / 0.6 = 0.9\). Thus, the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked is 0.9.

Calculate \(P(F \mid E)\)

Now, we calculate \(P(F \mid E)\) the probability that the second airline's flight is fully booked given that the first airline's flight is fully booked, following a similar method. We use the formula, \(P(F \mid E) = P(E ∩ F) / P(E)\), yielding: \(P(F \mid E) = 0.54 / 0.7 = 0.7714\). Thus, the probability that the second airline's flight is fully booked given that the first airline's flight is fully booked is 0.7714.