Consider a binomial distribution with 10 trials. Look at...

Consider a binomial distribution with 10 trials. Look at Table 3 (Appendix II) showing binomial probabilities for various values of $p$ the probability of success on a single trial. (a) For what value of $p$ is the distribution symmetric? What is the expected value of this distribution? Is the distribution centered over this value? (b) For small values of $p,$ is the distribution skewed right or left? (c) For large values of $p,$ is the distribution skewed right or left?

Consider a binomial distribution with 10 trials. Look at Table 3 (Appendix II) showing binomial probabilities for various values of $p$ the probability of success on a single trial.
(a) For what value of $p$ is the distribution symmetric? What is the expected value of this distribution? Is the distribution centered over this value?
(b) For small values of $p,$ is the distribution skewed right or left?
(c) For large values of $p,$ is the distribution skewed right or left?

Key Concepts

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It is characterized by the parameters n (the number of trials) and p (the probability of success on each trial), and it is used to model situations with two possible outcomes, such as success or failure.

Symmetry of the Distribution

A binomial distribution is symmetric when the probability of success, p, is 0.5. In this case, the distribution mirrors itself around the central value because the likelihood of success and failure are equal. This symmetry implies that the peak of the distribution (where the maximum probability exists) aligns with the mean, meaning the distribution is centered over its expected value.

Expected Value in a Binomial Distribution

The expected value, or mean, of a binomial distribution is given by np, 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 over many repetitions of the experiment and is a key measure of the distribution’s central tendency.

Skewness in Binomial Distributions

Skewness refers to the asymmetry of a probability distribution. For a binomial distribution, if p is less than 0.5 (small values of p), the distribution tends to be skewed to the right, meaning it has a longer tail on the right side. Conversely, when p is greater than 0.5 (large values of p), the distribution tends to be skewed to the left, meaning it has a longer tail on the left side. This skewness changes the visual balance of the distribution and its concentration around the mean.

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