Refer to the previous exercise. Suppose that 50% of the...

Refer to the previous exercise. Suppose that 50$\%$ of the overnight parcels are sent via express mail service $E_{2}$ and the remaining 10$\%$ are sent via $E_{3} .$ Of those sent via $E_{2},$ only 1$\%$ arrive late, whereas 5$\%$ of the parcels handled by $E_{3}$ arrive late. (a) What is the probability that a randomly selected parcel arrived late? (b) If a randomly selected parcel has arrived on time, what is the probability that it was not sent via $E_{1}$ ?

Refer to the previous exercise. Suppose that 50$\%$ of the overnight parcels are sent via express mail service $E_{2}$ and the remaining 10$\%$ are sent via $E_{3} .$ Of those sent via $E_{2},$ only 1$\%$ arrive late, whereas 5$\%$ of the parcels handled by $E_{3}$ arrive late.
(a) What is the probability that a randomly selected parcel arrived late?
(b) If a randomly selected parcel has arrived on time, what is the probability that it was not sent via $E_{1}$ ?

Key Concepts

Law of Total Probability

This principle states that the overall probability of an event can be computed by considering all the different ways that event can occur. It involves summing the probabilities of the event under each distinct condition, weighted by the probability of each condition occurring. This is particularly useful in situations where there are multiple mutually exclusive pathways leading to the same outcome.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It allows for refined probability calculations in scenarios where the outcome is dependent on prior events. In problems like this, computing the probability of an event under a specific condition, such as on-time delivery, is an application of conditional probability.

Bayes' Theorem

Bayes' Theorem provides a method to update the probability of a hypothesis based on new evidence. It relates the conditional and marginal probabilities of random events by 'inverting' the conditioning. This theorem is central to solving problems that require determining the probability that a given cause was responsible for an observed outcome.

Complementary Events

Complementary events are a pair of mutually exclusive events where one event is the 'opposite' of the other, and their probabilities add up to one. In many problems, knowing the probability of an event allows one to easily compute the probability of its complement. This concept is useful when calculating probabilities like 'not sent via E1' by considering the complement of being sent via E1.

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