You are investigating the punctuality of the airlines in...

You are investigating the punctuality of the airlines in Asia. Your survey tells you that, out of 15 airlines, $80 \%$ of them are likely to be late at least once a month. Assume the punctuality random variable follows a binomial distribution. Determine the following. a. Which assumptions do you need to make in order to be correct in considering a binomial distribution for your variable? b. How many airlines will be late in one month? c. What is the standard deviation of this random variable (i.e., the risk of being late)? d. What is the probability that they all will be late?

You are investigating the punctuality of the airlines in Asia. Your survey tells you that, out of 15 airlines, $80 \%$ of them are likely to be late at least once a month. Assume the punctuality random variable follows a binomial distribution. Determine the following.
a. Which assumptions do you need to make in order to be correct in considering a binomial distribution for your variable?
b. How many airlines will be late in one month?
c. What is the standard deviation of this random variable (i.e., the risk of being late)?
d. What is the probability that they all will be late?

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes—commonly referred to as success and failure—and where the probability of success remains constant across the trials.

Assumptions of Binomial Experiments

To appropriately use the binomial distribution, several assumptions must hold: there must be a fixed number of trials; each trial must be independent of the others; each trial should result in one of two possible outcomes (often labeled as success or failure); and the probability of success must be constant for all trials.

Expected Value and Variance in Binomial Distribution

The expected value (mean) of a binomial random variable is calculated by multiplying the number of trials (n) by the probability of success (p), which provides a measure of the central tendency. The variance, given by the product n * p * (1 - p), quantifies the spread or risk associated with the distribution, and the standard deviation is the square root of the variance.

Calculating Binomial Probabilities

The probability of observing exactly a specific number of successes in a series of binomial trials is determined using the formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k). This formula combines the number of ways to achieve k successes with the likelihood of each individual outcome occurring according to the success probability.

Transcript

Please give Ace some feedback