Question

a state lottery randomly chooses 6 balls numbered from 1 through 43 without replacement. you choose 6 numbers and purchase a lottery ticket. the random variable represents the number of matches on your ticket to the numbers drawn in the lottery. determine whether this experiment is binomial. if so, identify a success, specify the values n, p, and q and list the possible values of the random variable x.

is the experiment binomial?
a. no, there are more than two outcomes for each trial.
b. yes, there are a fixed number of trials and the trials are independent of each other.
c. yes, the probability of success is the same for each trial.
d. no, because the probability of success is different for each trial.

identify a success. choose the correct answer below.
a. a success is matching all of the numbers in the lottery.
b. a success is matching one of the numbers in the lottery.
c. a success is the number of matches on your ticket to the numbers drawn in the lottery.
d. the experiment is not binomial.

specify the values n, p, and q. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \\(n = \\), \\(p = \\), \\(q = \\)
(type integers or decimals.)
b. the experiment is not binomial.

list the possible values of the random variable x. choose the correct answer below.
a. the possible values for x are 1, 2, ..., 6.
b. the possible values for x are 1, 2, 3, 4, ..., 43.
c. the experiment is not binomial.

Explanation:

Analyze the binomial experiment conditions

The lottery selects 6 balls from 43 without replacement. Because the selection is without replacement, the probability of matching a number changes with each selection (the trials are dependent). Therefore, the experiment is not binomial.

$$ \text{Without replacement} \implies \text{Dependent trials} \implies \text{Not binomial} $$

Evaluate the first and second questions

• Is the experiment binomial?

The correct option is D: "No, because the probability of success is different for each trial."

• Identify a success.

Since the experiment is not binomial, we cannot define a standard binomial success.
The correct option is D: "The experiment is not binomial."

Evaluate the third and fourth questions

• Specify the values \(n\), \(p\), and \(q\).

The correct option is B: "The experiment is not binomial."

• List the possible values of the random variable \(x\).

The correct option is C: "The experiment is not binomial."

Answer:

Question 1

• (A) No, there are more than two outcomes for each trial.
• (B) Yes, there are a fixed number of trials and the trials are independent of each other.
• (C) Yes, the probability of success is the same for each trial.
• (D) No, because the probability of success is different for each trial. (Correct answer)

Question 2

• (A) A success is matching all of the numbers in the lottery.
• (B) A success is matching one of the numbers in the lottery.
• (C) A success is the number of matches on your ticket to the numbers drawn in the lottery.
• (D) The experiment is not binomial. (Correct answer)

Question 3

• (A) \(n = \square\), \(p = \square\), \(q = \square\)
• (B) The experiment is not binomial. (Correct answer)

Question 4

• (A) The possible values for x are 1, 2, ..., 6.
• (B) The possible values for x are 1, 2, 3, 4, ..., 43.
• (C) The experiment is not binomial. (Correct answer)