Delta International delivers approximately one million...

Delta International delivers approximately one million packages a day between East Asia and the United States. A random sample of the daily number of package delivery failures over the past six months provided the following results: $15,10,8,16,12,11,9,8$, $12,9,10,8,7,16,14,12,10,9,8,11$. There was nothing unusual about the operations during these days and, thus, the results can be considered typical. Using these data and your understanding of the delivery process answer the following: a. What probability model should be used and why? b. What is the probability of 10 or more failed deliveries on a typical future day? c. What is the probability of less than 6 failed deliveries? d. Find the number of failures such that the probability of exceeding this number is $10 \%$ or less.

Delta International delivers approximately one million packages a day between East Asia and the United States. A random sample of the daily number of package delivery failures over the past six months provided the following results: $15,10,8,16,12,11,9,8$, $12,9,10,8,7,16,14,12,10,9,8,11$. There was nothing unusual about the operations during these days and, thus, the results can be considered typical. Using these data and your understanding of the delivery process answer the following:
a. What probability model should be used and why?
b. What is the probability of 10 or more failed deliveries on a typical future day?
c. What is the probability of less than 6 failed deliveries?
d. Find the number of failures such that the probability of exceeding this number is $10 \%$ or less.

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a discrete distribution, such as the Poisson, is used to compute the probability that the random variable is less than or equal to a certain value. It involves summing up individual probabilities from zero to the specified value and is essential for probability queries that involve ranges, such as finding the probability of having at least or no more than a certain number of failures.

Quantile Computation

Quantile computation in the context of probability distributions involves finding the cut-off value below which a specified percentage of the distribution's probability mass lies. In practical terms, for the delivery failures model, this means determining a threshold such that there is only a specified low probability (e.g., 10%) of exceeding that threshold. This is useful for decision-making and risk assessment.

Parameter Estimation

Parameter estimation involves determining the distribution's parameter (lambda for the Poisson distribution) from observed data. This is often done by calculating the sample mean, which, under the Poisson assumption, is used as an unbiased estimate of lambda. This step is crucial for tailoring the theoretical model to the observed process.

Poisson Distribution

The Poisson distribution is used to model the number of times an event occurs within a fixed interval, provided that these occurrences are independent and the average rate (lambda) is constant. It is particularly appropriate for modeling rare or discrete events in time or space, such as delivery failures, when the likelihood of two events occurring simultaneously is negligible.

Transcript

00:01 In this problem, it is said that 95 % of first class mail within the same city is delivered within two days of the time of mailing.

00:09 Six letters are randomly sent to different locations and we need to answer a few questions.

00:15 So based on the information given in the question, let's consider x to be a random variable, which denotes the number of letters out of those six letters which arrive within two days of the time of mailing.

00:29 This random variable x is going to follow a binomial distribution.

00:35 In this binomial distribution, the parameters are n and p.

00:39 N is the number of trials, that will be six in this case because we're considering six letters, and p is the probability of success since 95 % of first class mail within the same city is delivered within two days of the time of mailing.

00:51 So p will be 95%, which is 95 over 100, or 0 .95.

00:57 Now recall the probability mass function for a binomial random variable x.

01:02 The probability of exactly r successes is given by ncr, p to the power of r, 1 minus p to the power of n minus r.

01:10 Now we're going to use this formula to find the answer to the first question, what is the probability that all six arrive within two days? so the probability of exactly six successes, we want r to be six in this case.

01:22 So we have ncr, p to the power of r, 1 minus p to the power of n minus r.

01:32 So if we calculate this, we'll see that this is approximately equal to 0 .73509.

01:38 So this is our required answer.

01:41 Now next, we want to find the probability that exactly five arrive within two days...

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