Questions: A state lottery randomly chooses 7 balls numbered from 1 through 39 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. Yes, there are a fixed number of trials and the trials are independent of each other. B. Yes, the probability of success is the same for each trial. C. No, there are more than two outcomes for each trial. D. No, because the probability of success is different for each trial.

Solution Steps

In this lottery experiment, you choose \( n = 7 \) numbers. Therefore, the fixed number of trials is:

\[ \text{Fixed number of trials} = n = 7 \]

The probability of matching a single number in the lottery is given by:

\[ p = \frac{1}{39} \approx 0.0256 \]

This indicates that the probability of success for each trial is approximately \( 0.0256 \).

In this experiment, the trials are not independent because the balls are drawn without replacement. As each ball is drawn, the total number of balls decreases, which alters the probability of success for subsequent trials. Thus, we conclude:

\[ \text{Are trials independent?} \quad \text{No, because the probability changes as balls are drawn without replacement.} \]

To determine if the experiment is binomial, we check the following conditions:

1. There is a fixed number of trials (\( n = 7 \)).
2. Each trial has only two outcomes (success or failure).
3. The probability of success is the same for each trial (not true here).
4. The trials are independent (not true here).

Since the probability of success changes with each draw, the experiment does not satisfy the conditions for a binomial distribution.

Final Answer