An expensive piece of equipment in a laboratory is starting to show signs of age. Let X be the number of days in any week that the equipment is working and suppose that X has the following probability distribution:

x	0	1	2	3	4	5
P(X=x)	0.01	0.09	0.25	0.34	0.24	0.07
In Quiz 3 we calculated the expected value and standard deviation of X.

The probabilities for X actually come from a Binomial distribution with n = 5 trials. The parameter p for this Binomial distribution is 

To find the parameter \( p \) of the Binomial distribution that matches the given probabilities, we'll use the properties of the Binomial distribution.

The Binomial distribution is defined as:

\[
P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}
\]

Given:
- \( n = 5 \)
- The probabilities \( P(X = x) \) for \( x = 0, 1, 2, 3, 4, 5 \) are provided.

To find \( p \), we should use the fact that for a Binomial distribution:

\[
\text{E}(X) = np
\]

We can calculate the expected value \( \text{E}(X) \) using the provided distribution and then set it equal to \( 5p \) (since \( n = 5 \)).

### Step 1: Calculate the Expected Value \( \text{E}(X) \)

\[
\text{E}(X) = \sum_{x=0}^{5} x \cdot P(X = x)
\]

Substituting the given probabilities:

\[
\text{E}(X) = 0 \cdot 0.01 + 1 \cdot 0.09 + 2 \cdot 0.25 + 3 \cdot 0.34 + 4 \cdot 0.24 + 5 \cdot 0.07
\]

\[
\text{E}(X) = 0 + 0.09 + 0.5 + 1.02 + 0.96 + 0.35 = 2.92
\]

### Step 2: Find \( p \)

Given \( \text{E}(X) = np \):

\[
2.92 = 5p
\]

\[
p = \frac{2.92}{5} = 0.584
\]

Thus, the parameter \( p \) for this Binomial distribution is approximately **0.584**.

Would you like more details on this process or have any questions?

Here are 5 related questions for further exploration:

1. How would the distribution change if the parameter \( p \) was slightly higher?
2. What is the variance of \( X \) for this Binomial distribution?
3. How does the Binomial distribution relate to the Normal distribution for large \( n \)?
4. How would you calculate the probability of the equipment working for exactly 4 days?
5. What does the standard deviation tell you about the reliability of the equipment?

**Tip:** When dealing with distributions, understanding both the mean and the variance is crucial for interpreting the behavior of random variables.

Learn how to calculate the parameter \( p \) for a Binomial distribution with given probabilities. Understand the concept of expected value and its application in probability theory. Suitable for advanced high school students and beyond.

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