Questions: A state lottery randomly chooses 7 balls numbered from 1 through 41 without replacement. You choose 7 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, the probability of success is the same for each trial. C. No, because the probability of success is different for each trial. D. Yes, there are a fixed number of trials and the trials are independent of each other.

A state lottery randomly chooses 7 balls numbered from 1 through 41 without replacement. You choose 7 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, the probability of success is the same for each trial. C. No, because the probability of success is different for each trial. D. Yes, there are a fixed number of trials and the trials are independent of each other.
Transcript text: A state lottery randomly chooses 7 balls numbered from 1 through 41 without replacement. You choose 7 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, the probability of success is the same for each trial. C. No, because the probability of success is different for each trial. D. Yes, there are a fixed number of trials and the trials are independent of each other.
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Solution

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Solution Steps

Step 1: Determine Binomial Suitability

For an experiment to be considered binomial, it must satisfy four conditions:

  1. The number of trials is fixed.
  2. Each trial has only two possible outcomes (success or failure).
  3. The probability of success is the same for each trial.
  4. The trials are independent. In the context of a lottery drawing without replacement, the experiment ^does not^ meet these criteria because the probability of success changes with each draw. Therefore, the lottery drawing experiment is ^not binomial^.
Step 2: Identifying a Success

A 'success' could hypothetically be defined as drawing a specific number. However, this definition conflicts with the requirement for the probability of success to remain constant across trials.

Step 3: Specifying $n, p, q$

In this non-binomial context, $n$ is the number of balls drawn. However, $p$ and $q$ cannot be consistently defined due to the changing probabilities with each draw. Thus, we cannot apply the binomial formula directly.

Step 4: Listing Possible Values of $x$

The possible values of $x$ range from 0 to $n$, representing the range of possible matches from none to all.

Final Answer:

Given the changing probability of success with each draw without replacement, the lottery drawing experiment ^cannot^ be accurately modeled as a binomial experiment.

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