Determine whether or not the random variable X is a binomial...

Determine whether or not the random variable $X$ is a binomial random variable. If so, give the values of $n$ and $p$. If not, explain why not. a. $X$ is the number of black marbles in a sample of 5 marbles drawn randomly and without replacement from a box that contains 25 white marbles and 15 black marbles. b. $X$ is the number of black marbles in a sample of 5 marbles drawn randomly and with replacement from a box that contains 25 white marbles and 15 black marbles. c. $X$ is the number of voters in favor of proposed law in a sample 1,200 randomly selected voters drawn from the entire electorate of a country in which $35 \%$ of the voters favor the law. d. $X$ is the number of fish of a particular species, among the next ten landed by a commercial fishing boat, that are more than 13 inches in length, when $17 \%$ of all such fish exceed 13 inches in length. e. $X$ is the number of coins that match at least one other coin when four coins are tossed at once.

Determine whether or not the random variable $X$ is a binomial random variable. If so, give the values of $n$ and $p$. If not, explain why not.
a. $X$ is the number of black marbles in a sample of 5 marbles drawn randomly and without replacement from a box that contains 25 white marbles and 15 black marbles.
b. $X$ is the number of black marbles in a sample of 5 marbles drawn randomly and with replacement from a box that contains 25 white marbles and 15 black marbles.
c. $X$ is the number of voters in favor of proposed law in a sample 1,200 randomly selected voters drawn from the entire electorate of a country in which $35 \%$ of the voters favor the law.
d. $X$ is the number of fish of a particular species, among the next ten landed by a commercial fishing boat, that are more than 13 inches in length, when $17 \%$ of all such fish exceed 13 inches in length.
e. $X$ is the number of coins that match at least one other coin when four coins are tossed at once.

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success in a single trial (p).

Independent Trials

Independence of trials is a key requirement for a binomial model, meaning that the outcome of any one trial does not affect the outcome of any other trial. This ensures that the probability of success remains constant throughout the experiment.

Fixed Number of Trials

A fundamental aspect of the binomial distribution is that the number of trials (n) must be predetermined and fixed in advance. The experiment must have a set number of repeated tests or observations.

Constant Probability of Success

For the binomial model, each trial must have the same probability (p) of success. The likelihood of a success should not change from one trial to the next, which is tied directly to the requirement that the trials are independent.

Sampling Methods – With and Without Replacement

The method of sampling significantly affects whether a binomial model is appropriate. Sampling with replacement generally preserves the independence and constant probability conditions required for a binomial experiment, whereas sampling without replacement can introduce dependence between trials, potentially violating the conditions necessary for a binomial distribution unless the population is very large relative to the sample.

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