In a state lottery, a player must choose 10 numbers between 1 and 49, inclusive. On drawing day, 4 balls numbered between 1 and 49 are then randomly selected from an urn. The random variable [tex]$X$[/tex] represents the number of matching numbers. Verify that this satisfies the criteria for a hypergeometric probability experiment and determine the values for [tex]$N , n$[/tex], and k. Also list the possible values of the random variable X.
Is this a hypergeometric probability experiment?
A. No, because the population does not have finite size.
B. No, because there are not exactly two possible outcomes for each trial.
C. Yes, because the population has finite size, there are exactly two possible outcomes for each trial, and the sample is obtained from the population without replacement.
D. No, because the sample is not obtained from the population without replacement.
What are the values of [tex]$N , n$[/tex], and k?
[tex]$\begin{array}{l}
N=\square \\
n=\square \\
k=\square
\end{array}$[/tex]
What is the range of possible values for the random variable [tex]$X$[/tex]?
[tex]$\square$[/tex]
Answer (2)
The problem describes a lottery where a player picks 10 numbers out of 49, and 4 numbers are drawn.
The scenario fits the criteria for a hypergeometric experiment: finite population, two outcomes, and no replacement.
The parameters are identified as: N = 49 (population size), n = 4 (sample size), and k = 10 (number of successes in the population).
The possible values for the random variable X (number of matches) are 0, 1, 2, 3, and 4, since you can have at most 4 matches. 0 , 1 , 2 , 3 , 4
Explanation
Analyze the problem Let's analyze the problem. We are given a lottery scenario where a player chooses 10 numbers out of 49. Then, 4 numbers are drawn. We want to determine if this is a hypergeometric experiment, find the values of N, n, and k, and list the possible values of the random variable X, which represents the number of matching numbers.
Verify hypergeometric experiment criteria A hypergeometric experiment has the following characteristics:
A population of finite size.
Each draw has two possible outcomes: success (a match) or failure (no match).
The sample is drawn without replacement.
In this case:
The population size is finite (49 numbers).
Each number drawn either matches one of the player's numbers (success) or it doesn't (failure).
The numbers are drawn without replacement.
Therefore, this is a hypergeometric experiment.
Determine N, n, and k Now, let's identify the values of N, n, and k:
N is the population size, which is the total number of numbers to choose from: N = 49 .
n is the sample size, which is the number of balls drawn: n = 4 .
k is the number of successes in the population, which is the number of numbers the player chooses: k = 10 .
Determine the range of X The random variable X represents the number of matching numbers. The possible values of X range from 0 (no matches) to the smaller of n and k , which is min ( 4 , 10 ) = 4 . Therefore, the possible values of X are 0, 1, 2, 3, and 4.
Final Answer Based on the analysis, the experiment is hypergeometric. The values are N = 49 , n = 4 , and k = 10 . The possible values for the random variable X are 0, 1, 2, 3, and 4.
Examples
Hypergeometric distributions are useful in quality control. For example, suppose a batch of 50 items contains 10 defective items. If you randomly select 5 items from the batch, the hypergeometric distribution can be used to calculate the probability of finding a certain number of defective items in your sample. This helps determine if the batch meets quality standards.
The lottery scenario meets the criteria for a hypergeometric experiment, with values N = 49, n = 4, and k = 10. The possible values for the matching random variable X range from 0 to 4. Therefore, the correct multiple choice option is C.
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