Problem 204 - Applied Statistics and Probability for Engineers

Question

The probability that a customer's order is not shipped on time is \(0.05 .\) A particular customer places three orders, and the orders are placed far enough apart in time that they can be considered to be independent events. (a) What is the probability that all are shipped on time? (b) What is the probability that exactly one is not shipped on time? (c) What is the probability that two or more orders are not shipped on time?

Step-by-Step Solution

Answer

(a) 0.857375, (b) 0.135375, (c) 0.00725

Step 1: Define the Probability of On-time Shipping

The probability that an order is not shipped on time is given as \(P( ext{not on time}) = 0.05\). Therefore, the probability that an order is shipped on time is \(P( ext{on time}) = 1 - 0.05 = 0.95\).

Step 2: Probability All Orders Shipped on Time

For three independent orders, the probability that all are shipped on time is the product of the probabilities of each being on time: \(P( ext{all on time}) = 0.95 \times 0.95 \times 0.95 = 0.95^3\). Calculate \(0.95^3 = 0.857375\).

Step 3: Probability Exactly One is Not Shipped on Time

To find the probability that exactly one order out of three is not shipped on time, we use the binomial probability formula. Let \(X\) be the number of orders not shipped on time: \(P(X = 1) = \binom{3}{1} (0.05)^1 (0.95)^2\). Calculate: \(\binom{3}{1} = 3\), and thus \(3 \times 0.05 \times 0.95^2 = 3 \times 0.05 \times 0.9025 = 0.135375\).

Step 4: Probability of Two or More Not Shipped on Time

The probability of two or more orders not shipped on time is the complement of the probabilities of 0 or 1 order not shipped on time: \(P(X \geq 2) = 1 - (P(X = 0) + P(X = 1))\). From Step 2, \(P(X = 0) = 0.857375\). From Step 3, \(P(X = 1) = 0.135375\). Thus, \(P(X \geq 2) = 1 - (0.857375 + 0.135375) = 0.00725\).

Key Concepts

Binomial Distribution

The binomial distribution is a powerful tool in probability theory that deals with scenarios where there are two possible outcomes for each trial. These outcomes are typically classified as "success" and "failure". In the context of our exercise, the success is an order being shipped on time, while a failure means the order isn't shipped on time.

Binomial distribution is characterized by:

• The number of trials, denoted as \( n \)
• The probability of success in a single trial, denoted as \( p \)
• The probability of failure is \( 1 - p \)

To determine probabilities in a binomial distribution, we use the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) represents the binomial coefficient, calculated as the number of ways \( k \) successes can occur in \( n \) trials. This formula is especially useful when calculating the likelihood of a specific number of occurrences, like one order not being shipped on time in our problem.

Independent Events

In probability, independent events refer to scenarios where the outcome of one event does not influence the outcome of another. For example, when tossing a coin, the result of any single flip doesn't affect another.

In our exercise, each order's shipping time is an independent event. This means the probability that one order is shipped on time doesn't change the probability for another.

To determine the probability of multiple independent events all occurring, we simply multiply their individual probabilities together. In this exercise, since the probability of one order being shipped on time is \(0.95\), the probability of all three orders being shipped on time is given by:

• \(0.95 \times 0.95 \times 0.95 = 0.857375\)

Applying this principle helps us easily compute complex probabilistic scenarios when events are independent.

Complementary Probabilities

The concept of complementary probabilities is integral in understanding various probability scenarios. Complementary events are pairs of outcomes where one event occurring means the other cannot and vice versa.

For any event, its complement represents the rest of all possible outcomes:

• If an event has a probability \( P(A) \), then its complement has probability \( 1 - P(A) \)

In the order shipping problem, if the probability of an order being shipped late is 0.05, then the complementary probability, which is being shipped on time, is 0.95.

This approach also comes in handy when calculating probabilities like two or more orders not being shipped on time. We compute complementary probabilities to simplify calculations, subtracting the known possibilities from 1. This allows for quick computation of desired probabilities, like determining that two or more orders are not shipped on time using: \[P(X \geq 2) = 1 - (P(X = 0) + P(X = 1))\].

Order Shipment Probability

Calculating order shipment probability is a practical application of probability theory principles, especially useful in logistics and supply chain management.

In this problem, having the probability of an individual order not being shipped on time as \(0.05\) allows us to use this information to calculate multiple related probabilities:

• The likelihood of all orders being on time
• Exactly one being late
• Two or more being late

By applying the probability of shipping on time to independent events, we easily calculated the probability of all orders being on time. We used the binomial distribution formula to find exactly one out of three orders being late, taking advantage of independent events and their respective probabilities. Lastly, through complementary probabilities, we determined the chance of two or more orders being late by considering the combinations of other available outcomes.

This scenario helps students apply multiple probability theory concepts to real-world situations, improving understanding and problem-solving skills.