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In the world of statistics, probability is a crucial concept that helps us quantify the likelihood of different outcomes. In our mail delivery problem, we are dealing with a binomial distribution. Each letter's delivery within 2 days is a simple yes or no, i.e., success or failure. The probability of success in each trial (each letter) is given as 0.95.

A binomial distribution is used when there are a fixed number of independent trials (like the 6 letters), each with the same probability of success. We calculate specific probabilities using the binomial probability formula:

• The probability that all six letters arrive within 2 days is calculated as \[P(X = 6) = \binom{6}{6} (0.95)^6 (0.05)^0 = (0.95)^6 \]. This equals approximately 0.7351, meaning there's a 73.51% chance all will arrive on time.
• If we want precisely five letters to succeed in arrival on time, the formula is \[P(X = 5) = \binom{6}{5} (0.95)^5 (0.05)^1\]. This probability is approximately 0.2322 or 23.22%.

These calculations show how likely different outcomes are, providing a complete picture of what we can expect on average.

Another key aspect of the binomial distribution involves understanding the mean and variance.

The **mean** (often represented as \( \mu \)) provides the average expected outcome. For our mail scenario, the mean is calculated using the formula:

• \(\mu = n \times p\)

where \( n = 6 \) and \( p = 0.95 \). Hence, the mean number of letters expected to arrive on time is 5.7.This suggests that, on average, about 5 or 6 letters will arrive within the 2-day period.

The **variance** measures how much the results will vary from this mean on average. It's calculated as:

• \(\sigma^2 = n \times p \times (1-p)\)

Substituting the values, we have \( 6 \times 0.95 \times 0.05 = 0.285 \). This tells us about the distribution of numbers around the mean.

Understanding these concepts helps us anticipate variability and set expectations about delivery performance.

Standard deviation is another measure closely related to variance. While variance provides a measure of how data is spread around the mean, standard deviation gives us a direct and interpretable figure. It represents the average amount each number in a distribution differs from the mean.

Calculating the standard deviation for our binomially distributed mail scenario involves taking the square root of the variance. If the variance \(\sigma^2\) is 0.285, then the standard deviation \(\sigma\) is calculated as:

• \(\sigma = \sqrt{0.285} \approx 0.5347\)

This result shows that the number of letters that might deviate from the mean (5.7 letters) is roughly 0.53.

In practical terms, understanding the standard deviation allows us to expect some variability, but still provides confidence in the average delivery scenario. It's a valuable tool for predicting outcomes in statistical practices and forms the foundation of understanding data distribution patterns.