Questions: A state lottery randomly chooses 6 balls numbered from 1 through 41 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.
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, ..., 41.
C. The experiment is not binomial.
Transcript text: A state lottery randomly chooses 6 balls numbered from 1 through 41 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 $\mathrm{n}, \mathrm{p}$, and $\mathrm{q}$ and list the possible values of the random variable $x$.
Specify the values $\mathrm{n}, \mathrm{p}$, and $\mathrm{q}$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $\mathrm{n}=$ $\square$ . $\mathrm{p}=$ $\square$ . $q=$ $\square$
(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 $1,2, \ldots, 6$.
B. The possible values for $x$ are $1,2,3,4, \ldots, 41$.
C. The experiment is not binomial.
Solution
Solution Steps
Step 1: Determine the Nature of the Experiment
The lottery experiment involves drawing 6 balls from a total of 41 without replacement. This means that the trials are not independent, and the probability of success changes with each draw. Therefore, the experiment does not satisfy the conditions for a binomial distribution.
Step 2: Identify the Distribution
Since the trials are not independent and the probability of success varies, the appropriate distribution for this scenario is the hypergeometric distribution.
Step 3: Calculate the Probability of Matches
The probability of obtaining exactly \( k \) matches (where \( k \) can take values from 0 to 6) is given by the formula:
\[
P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}
\]
where:
\( N = 41 \) (total number of balls),
\( K = 6 \) (number of balls you choose),
\( n = 6 \) (number of balls drawn),
\( k \) is the number of matches.
The calculated probabilities for each value of \( k \) are as follows:
\( P(X = 0) = 0.361 \)
\( P(X = 1) = 0.4332 \)
\( P(X = 2) = 0.1747 \)
\( P(X = 3) = 0.0291 \)
\( P(X = 4) = 0.002 \)
\( P(X = 5) = 0.0 \)
\( P(X = 6) = 0.0 \)
Step 4: Calculate the Mean, Variance, and Standard Deviation
The mean \( \mu \), variance \( \sigma^2 \), and standard deviation \( \sigma \) of the hypergeometric distribution are calculated as follows: