Math Problem Statement
An electronics store has received a shipment of 25 table radios that have connections for an iPod or iPhone. Ten of these have two slots (so they can accommodate both devices), and the other fifteen have a single slot. Suppose that five of the 25 radios are randomly selected to be stored under a shelf where the radios are displayed, and the remaining ones are placed in a storeroom. Let X = the number among the radios stored under the display shelf that have two slots.
(a) What kind of distribution does X have (name and values of all parameters)?
Solution
This problem involves a situation where you are selecting 5 radios from a total of 25 radios, of which 10 have two slots and 15 have only one slot. The random variable X represents the number of radios with two slots among the 5 radios selected to be placed under the display shelf.
Distribution Type:
The situation described is one of sampling without replacement, as once a radio is selected, it is not returned to the pool of 25 radios for further selections. This suggests that X follows a hypergeometric distribution.
Parameters of the Hypergeometric Distribution:
The hypergeometric distribution is characterized by three parameters:
- : The total number of items (radios), which is 25.
- : The number of items that are of interest (radios with two slots), which is 10.
- : The number of items sampled (radios placed under the shelf), which is 5.
Thus, X follows a hypergeometric distribution with parameters:
Would you like me to calculate probabilities or other details based on this distribution?
Further related questions:
- What is the probability that exactly 2 of the radios under the shelf have two slots?
- What is the probability that all 5 radios under the shelf have two slots?
- What is the expected number of radios under the shelf that have two slots?
- What is the variance of the number of radios under the shelf that have two slots?
- How does the distribution change if we instead select 6 radios to place under the shelf?
Tip:
The hypergeometric distribution is useful in scenarios where you're sampling without replacement, which differs from the binomial distribution that assumes replacement (independent trials).
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Math Problem Analysis
Mathematical Concepts
Probability
Hypergeometric Distribution
Formulas
Hypergeometric distribution: X ~ Hypergeometric(N, K, n)
N: total number of items, K: number of successes, n: number of draws
Theorems
Hypergeometric distribution theorem: This applies to sampling without replacement, where the probability of each sample depends on the previous selection.
Suitable Grade Level
Grades 10-12
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