Question

Leah is flying from Boston to Denver with a connection in Chicago. The probability her first flight leaves on time is \(0.15\). If the flight is on time, the probability that her luggage will make the connecting flight in Chicago is \(0.95\), but if the first flight is delayed, the probability that the luggage will make it is only \(0.65\). a) Are the first flight leaving on time and the luggage making the connection independent events? Explain. b) What is the probability that her luggage arrives in Denver with her?

Short answer

a) No, the events are not independent. b) Probability is 0.7325.

Step-by-step solution

Understanding Independence

Two events, say A and B, are independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, events A and B are independent if \( P(A \cap B) = P(A) \cdot P(B) \). In this case, event A is the first flight leaving on time and B is the luggage making the connection.

Determine Luggage Connection Formula

Calculate \( P(Luggage) \) if first flight is on time: \( P(Luggage | On\ Time) = 0.95 \).Calculate \( P(Luggage) \) if flight is delayed: \( P(Luggage | Delayed) = 0.65 \).

Calculate Joint Probability

For the events to be independent, \( P(On\ Time \cap Luggage) = P(On\ Time) \cdot P(Luggage) \). Calculate \( P(On\ Time \cap Luggage) = P(On\ Time) \cdot P(Luggage | On\ Time) = 0.15 \cdot 0.95 = 0.1425 \).

Calculate Total Probability of Luggage

Calculate \( P(Luggage) = P(On\ Time) \cdot P(Luggage | On\ Time) + P(Delayed) \cdot P(Luggage | Delayed) \). Since \( P(Delayed) = 1 - 0.15 = 0.85 \), Thus, \( P(Luggage) = 0.15 \cdot 0.95 + 0.85 \cdot 0.65 = 0.7325 \).

Evaluate Independence

We calculated \( P(On\ Time \cap Luggage) = 0.1425 \) and \( P(On\ Time) \cdot P(Luggage) = 0.15 \cdot 0.7325 = 0.109875 \). Since these probabilities differ, the events "First flight on time" and "Luggage makes connection" are not independent.

Total Probability of Luggage Arriving

From Step 4, \( P(Luggage) = 0.7325 \). Since we need the probability that her luggage arrives in Denver with her, we are interested in the probability of both connection and time events occurring. This does not require further adjustments as \( P(Luggage) \) already considers both conditions.

Key concepts

Independent Events

When we talk about independent events in probability, we're exploring a situation where the outcome of one event does not influence the outcome of another. This is crucial for determining whether events impact each other's likelihood. If two events, A and B, are independent, the probability of them both occurring, denoted as \(P(A \cap B)\), is equal to the product of their individual probabilities, \(P(A) \cdot P(B)\).

In Leah's example, let's assign:

• Event A: The first flight leaves on time (probability of 0.15).
• Event B: The luggage makes the connecting flight.

In her case, we computed the joint probability \(P(A \cap B)\) to be 0.1425. If these events were independent, this joint probability should have equaled the product of \(P(A)\) times \(P(B)\), which was calculated as 0.109875. These two numbers are different, showing that whether or not the first flight is on time does affect the luggage connection. Thus, they are not independent events.

Conditional Probability

Conditional probability is a way of expressing the probability of one event occurring in the context of another event having already occurred. It's like asking, "Given that this has happened, what are the chances that this will also happen?" In mathematical terms, the conditional probability of event B given event A is denoted as \(P(B|A)\).

In our exercise, given the first flight leaves on time, we know the probability that Leah's luggage makes the connection in Chicago is 0.95. This is written as \(P(Luggage | On\ Time) = 0.95\). Alternatively, if her first flight is delayed, this conditional probability drops to 0.65, noted as \(P(Luggage | Delayed) = 0.65\).

By analyzing these numbers, you see how the condition — whether the flight is on time or not — changes the probability of her luggage making the connection. Understanding conditional probability helps us predict outcomes more accurately when we have some prior information.

Total Probability Theorem

The Total Probability Theorem is a fundamental rule that helps transform complex probability scenarios into simpler, more manageable parts. This theorem is especially handy when you want to find the probability of an event based on several other events which cover all possible outcomes.

In Leah's journey, we're interested in the probability that her luggage makes it through to Denver. We denote this as \(P(Luggage)\), which depends on whether her first flight is on time or delayed. The Total Probability Theorem lets us calculate \(P(Luggage)\) like this:

• Calculate the probability her luggage makes the connection if her flight is on time: \(0.15 \cdot 0.95 = 0.1425\).
• Calculate it if her flight is delayed: \(0.85 \cdot 0.65 = 0.5525\).
• Add these probabilities together to get the total probability: \(0.1425 + 0.5525 = 0.7325\).

The Total Probability Theorem shows how each possible scenario (on time or delayed) contributes to the overall likelihood of her luggage arriving. It's a systematic way to sum across all potential paths to a final outcome.

Joint Probability

Joint probability refers to the likelihood of two events happening at the same time. It's an intriguing look at how events intersect in complex scenarios. In the notation \(P(A \cap B)\), it signifies the probability that both A and B occur. In analyses, joint probabilities reveal intersections of events and are foundational in computing other probabilities like conditional probabilities.

In Leah's situation, we want to find the probability that her first flight is on time and her luggage makes the connection in Chicago. This is a classic joint probability scenario. From our calculations, \(P(On\ Time \cap Luggage) = 0.1425\).

Joint probability helps us understand scenarios where two potentially related events both take place. It gives us insight into whether or not events are independent. In Leah's case, it clarifies how the timing of her flight impacts the delivery of her luggage.