Question

franco has a hat with 9 blue marbles, 4 yellow marbles, and 12 green marbles. he designs a binomial experiment by drawing a marble from the hat, recording whether the marble is green, and then laying the marble aside. he then repeats the process seven times. which statement must be true? the experiment is not a binomial experiment because there is not a fixed number of drawings. the experiment is not a binomial experiment because the probability of choosing a green marble is the same for each drawing. the experiment is not a binomial experiment because the probability of choosing a green marble is not the same for each drawing. the experiment is not a binomial experiment because there are only two possible outcomes for each drawing.

Explanation:

Step1: Recall binomial experiment criteria

A binomial experiment has a fixed number of trials, independent trials, two - possible outcomes per trial, and a constant probability of success on each trial.

Step2: Analyze the given experiment

The number of trials is fixed ($n = 7$). There are two possible outcomes (green or not - green). But since the marbles are laid aside (sampling without replacement), the probability of choosing a green marble changes for each drawing. For example, the first - time probability of choosing a green marble is $P_1=\frac{12}{9 + 4+12}=\frac{12}{25}$, and the second - time probability depends on whether a green marble was chosen the first time. If a green marble was chosen first, the second - time probability is $P_2=\frac{11}{24}$, and if a non - green marble was chosen first, the second - time probability is $P_2=\frac{12}{24}$.

Answer:

The experiment is not a binomial experiment because the probability of choosing a green marble is not the same for each drawing.