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)?

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:
1. \( N \): The total number of items (radios), which is 25.
2. \( K \): The number of items that are of interest (radios with two slots), which is 10.
3. \( n \): The number of items sampled (radios placed under the shelf), which is 5.

Thus, *X* follows a hypergeometric distribution with parameters:

\[
X \sim \text{Hypergeometric}(N = 25, K = 10, n = 5)
\]

Would you like me to calculate probabilities or other details based on this distribution?

### Further related questions:
1. What is the probability that exactly 2 of the radios under the shelf have two slots?
2. What is the probability that all 5 radios under the shelf have two slots?
3. What is the expected number of radios under the shelf that have two slots?
4. What is the variance of the number of radios under the shelf that have two slots?
5. 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).

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)?

Explore the hypergeometric distribution with a problem involving 25 table radios, 10 of which have two slots. Understand how to calculate the probability distribution when sampling without replacement, and learn the formula and parameters for this probability scenario. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation