The United States Postal Service reports 95 percent of first...

The United States Postal Service reports 95 percent of first class mail within the same city is delivered within two days of the time of mailing. Six letters are randomly sent to different locations. a. What is the probability that all six arrive within two days? b. What is the probability that exactly five arrive within two days? c. Find the mean number of letters that will arrive within two days. d. Compute the variance and standard deviation of the number that will arrive within two days.

The United States Postal Service reports 95 percent of first class mail within the same city is delivered within two days of the time of mailing. Six letters are randomly sent to different locations.
a. What is the probability that all six arrive within two days?
b. What is the probability that exactly five arrive within two days?
c. Find the mean number of letters that will arrive within two days.
d. Compute the variance and standard deviation of the number that will arrive within two days.

Key Concepts

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has exactly two outcomes (commonly labeled as 'success' and 'failure') and each trial has the same probability of success. This distribution is useful when analyzing events that have a binary outcome, such as whether a letter is delivered within a certain time frame.

Binomial Probability Formula

The binomial probability formula is used to calculate the probability of obtaining exactly k successes in n independent Bernoulli trials. The formula is given by P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success and (n choose k) represents the number of ways to choose k successes from n trials.

Expected Value (Mean) of a Binomial Distribution

The expected value or mean of a binomial random variable is calculated by multiplying the total number of trials by the probability of success, n * p. This measure gives the average number of successes one would expect over the long run.

Variance and Standard Deviation of a Binomial Distribution

The variance of a binomial distribution measures the spread of the distribution and is computed as n * p * (1-p), where p is the probability of success. The standard deviation, which is the square root of the variance, provides a measure of the average deviation from the mean, indicating how much the number of successes typically fluctuates around the expected value.

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