Final Answer:
It appears that the question is incomplete and missing some vital information, specifically the probability value 'p' for the success of finding the desired type of radio among those stored under the display shelf. However, we can still discuss the nature of the distribution given the information provided, which is that the random variable `X` represents the number of radios with two slots.
Step-by-step explanation:
Given the information, we can infer the following about the distribution:
1. It is binomial — because the problem seems to involve a fixed number of independent trials (the number of radios checked), with each trial having two possible outcomes: a radio either has two slots (success) or it doesn't (failure).
2. The number of trials, denoted as \( n \), is given as 25. This suggests that there are 25 radios to check under the display shelf.
3. The number of successful outcomes we are interested in, denoted as \( x \), is given as 10. This would mean we are looking for the probability of finding exactly 10 radios with two slots out of the 25 checked.
4. The probability of success on a single trial, denoted as \( p \), is not given in the question. This value is critical because it defines the likelihood that any one radio has two slots. Without this value, we cannot calculate the probability distribution for \( X \).
To make this a complete binomial distribution, we need the value of \( p \). The binomial distribution would then be defined by the parameters \( n = 25 \), \( x = 10 \), and \( p \), where \( p \) is the probability of a radio having two slots.
Once \( p \) is known, we could compute the probability of finding exactly 10 radios with two slots using the binomial probability formula:
\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot (1 - p)^{n - x} \]
Where:
- \( \binom{n}{x} \) is the binomial coefficient that calculates the number of ways to choose \( x \) successes out of \( n \) trials.
- \( p^x \) is the probability of having \( x \) successes.
- \( (1 - p)^{n - x} \) is the probability of having \( n - x \) failures.
In conclusion, the random variable \( X \) follows a binomial distribution with \( n = 25 \) and \( x = 10 \), but without the value of \( p \), we cannot fully define the distribution or calculate specific probabilities. If you provide the value of \( p \), we can then proceed with the calculation.