The U.S. Postal Service reports 95 % of first-class mail...

The U.S. Postal Service reports $95 \%$ of first-class mail within the same city is delivered within 2 days of the time of mailing. Six letters are randomly sent to different locations. a. What is the probability that all six arrive within 2 days? b. What is the probability that exactly five arrive within 2 days? c. Find the mean number of letters that will arrive within 2 days. d. Compute the variance and standard deviation of the number that will arrive within 2 days.

The U.S. Postal Service reports $95 \%$ of first-class mail within the same city is delivered within 2 days of the time of mailing. Six letters are randomly sent to different locations.
a. What is the probability that all six arrive within 2 days?
b. What is the probability that exactly five arrive within 2 days?
c. Find the mean number of letters that will arrive within 2 days.
d. Compute the variance and standard deviation of the number that will arrive within 2 days.

Key Concepts

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) and a constant probability of success. This framework is ideal for problems where events (such as letter delivery times) occur under identical conditions and are statistically independent.

Independent Trials

The assumption of independent trials means that the outcome of one trial does not affect the outcome of any other trial. When applying the binomial distribution, each trial (in this case, the delivery of a letter) is considered separately, ensuring that the probability of success remains constant for each trial.

Probability Mass Function (PMF) of a Binomial Random Variable

The PMF of a binomial random variable is given by the formula P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where n is the number of trials, k is the number of successes, and p is the probability of success on each trial. This function is used to calculate the probability of exactly k successes (like the number of letters arriving on time) in n trials.

Expected Value of a Binomial Distribution

The expected value (mean) of a binomial distribution is calculated as ? = n * p, where n is the number of trials and p is the probability of success on each trial. This value represents the average number of successes one would expect over many repetitions of the experiment.

Variance and Standard Deviation of a Binomial Distribution

The variance of a binomial distribution is calculated using the formula ?² = n * p * (1 - p), and the standard deviation is the square root of the variance. These measures indicate the spread or variability of the number of successes around the mean, describing how much the number of successes is expected to vary from the average.

Transcript

00:01 All right, so we're looking at the post office, and they're doing a little binomial experiment here.

00:07 95 % of first class mail will arrive within two days at the time of mailing, and they're testing it with six letters and seeing what's the probability about some numbers arriving within two days.

00:21 And looking at variance in standard deviation.

00:23 So let's go and do that.

00:25 So it's binomial because we know that there's a fixed, you have a six, or a failure, the probability of success and failure is the same, and it's a very clearly defined number of trials that we have.

00:43 So i'm going to use the spreadsheet to do our work here.

00:49 So we're going to use this formula here equals combination of n.

00:56 Oh, i should define our n.

00:58 N is six because there are six letters.

01:01 And probability is 0 .95 .95.

01:08 So equals combination of 6 choose x, which is that, 6 choose the part of the x value we want, times the probability to the x power times 1 minus the probability to the n, which is 6, n which is six minus x power right and if we've done this correct which i think we have it should sum to one it's always a double check that does which means it's a probability distribution so that's good but good uh so now answer for a what's the probability that all six will arrive within two days point seven three five right there what's the probability 0 .735.

02:13 What's the probability that exactly 5? right there.

02:16 0 .232.

02:19 What's the mean number of letters? so the mean of a binomial distribution is n times pi...

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