Choose the logical binomial random variable.

Multiple choice question.

The number of late shipments out of the next 12 sent by one company.


The number of late shipments over the next month.


The number of late shipments and missing shipments out of the next 12 sent.


The number of late shipments for two different companies.

To choose the logical binomial random variable, we need to identify the scenario that fits the criteria of a binomial distribution. A binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. The key features are:

1. A fixed number of trials (\(n\)).
2. Each trial has two possible outcomes (success or failure).
3. The probability of success (\(p\)) is constant for each trial.
4. The trials are independent.

Given these criteria, let's analyze each option:

1. **The number of late shipments out of the next 12 sent by one company.**
   - This option fits a binomial distribution because there is a fixed number of trials (12 shipments), each shipment can be either late (success) or not late (failure), and the probability of being late is constant.

2. **The number of late shipments over the next month.**
   - This option does not fit a binomial distribution as it does not specify a fixed number of trials. The number of shipments in a month can vary.

3. **The number of late shipments and missing shipments out of the next 12 sent.**
   - This option does not fit a binomial distribution because it involves more than two outcomes (late, missing, or on time).

4. **The number of late shipments for two different companies.**
   - This option does not fit a binomial distribution because it involves two different sets of trials, which may have different probabilities and are not necessarily independent.

Therefore, the correct answer is:

**The number of late shipments out of the next 12 sent by one company.**

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How would you calculate the probability of exactly 3 late shipments out of the next 12?
2. What is the mean number of late shipments if the probability of a late shipment is 0.1?
3. How does the independence of trials affect the binomial distribution?
4. What is the variance of the number of late shipments out of the next 12 if the probability of being late is 0.2?
5. Can the binomial distribution be used if the probability of success changes for each trial?
6. What are some real-world examples of binomial distributions?
7. How would you use the binomial formula to find the probability of no late shipments out of the next 12?
8. How does the binomial distribution differ from other probability distributions, such as the Poisson distribution?

**Tip:** When identifying a binomial random variable, always check for a fixed number of trials, two possible outcomes, a constant probability of success, and the independence of trials.

Learn how to choose the logical binomial random variable with this example problem involving late shipments. Understand the characteristics of binomial distributions, such as fixed number of trials, two possible outcomes, constant probability of success, and independence of trials. Suitable for high school students and anyone interested in probability theory.

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