**Exercise 6.30: Probability Analysis of Flight Bookings**Two different airlines offer a flight from Los Angeles to New York that departs each weekday morning at the same scheduled time. Define event $ E $ as the occurrence of the first airline's flight being fully booked on a specific day, and event $ F $ as the occurrence of the second airline's flight being fully booked on the same day.Given the probabilities:- $ P(E) = 0.7 $- $ P(F) = 0.6 $- $ P(E \cap F) = 0.54 $**Tasks:**a. Calculate $ P(E|F) $, the conditional probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. (Hint: Refer to Example 6.12 for guidance.)b. Calculate $ P(F|E) $, the conditional probability that the second airline’s flight is fully booked given that the first airline’s flight is fully booked. Consider using the formula for conditional probability:\[ P(E|F) = \frac{P(E \cap F)}{P(F)} \]\[ P(F|E) = \frac{P(E \cap F)}{P(E)} \]

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6.30. 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(EN F) = .54. a. Calculate P(E\F), the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. (Hint: See Example 6.12.) b. Calculate P(E|F).

6.30. 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(EN F) = .54. a. Calculate P(E\F), the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. (Hint: See Example 6.12.) b. Calculate P(E|F).

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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**Exercise 6.30: Probability Analysis of Flight Bookings**

Two different airlines offer a flight from Los Angeles to New York that departs each weekday morning at the same scheduled time. Define event $ E $ as the occurrence of the first airline's flight being fully booked on a specific day, and event $ F $ as the occurrence of the second airline's flight being fully booked on the same day.

Given the probabilities:
- $ P(E) = 0.7 $
- $ P(F) = 0.6 $
- $ P(E \cap F) = 0.54 $

**Tasks:**

a. Calculate $ P(E|F) $, the conditional probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. (Hint: Refer to Example 6.12 for guidance.)

b. Calculate $ P(F|E) $, the conditional probability that the second airline’s flight is fully booked given that the first airline’s flight is fully booked.

Consider using the formula for conditional probability:
\[ P(E|F) = \frac{P(E \cap F)}{P(F)} \]
\[ P(F|E) = \frac{P(E \cap F)}{P(E)} \]

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