Let's go step-by-step through the problem.

1. Probability Distribution

We are given the probabilities for 0, 1, 2, and 3 medical tests. Here's the probability distribution:

Medical Tests $x$	Probability $P(x)$
0	$\frac{6}{15}$
1	$\frac{5}{15}$
2	$\frac{3}{15}$
3	$\frac{1}{15}$

2. What is the probability of 1 medical test being performed?

The probability of exactly 1 test being performed is: $P(x = 1) = \frac{5}{15} = \frac{1}{3}$

3. What is the probability of less than 2 medical tests being performed?

To find $P(x < 2)$, we sum the probabilities for 0 and 1 medical tests: $P(x < 2) = P(x = 0) + P(x = 1) = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$

4. What is the probability of between 0 and 1 tests being performed, inclusive?

To find $P(0 \leq x \leq 1)$, sum the probabilities for 0 and 1 medical tests: $P(0 \leq x \leq 1) = P(x = 0) + P(x = 1) = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$

5. Mean ( $\mu$ ) and Standard Deviation ( $\sigma$ )

To find the mean and standard deviation, we will use the formulas:

• Mean: $\mu = \sum x P(x)$ Plugging in the values: $\mu = 0 \times \frac{6}{15} + 1 \times \frac{5}{15} + 2 \times \frac{3}{15} + 3 \times \frac{1}{15} = 0 + \frac{5}{15} + \frac{6}{15} + \frac{3}{15} = \frac{14}{15} \approx 0.933$

• Standard deviation: $\sigma = \sqrt{\sum (x - \mu)^2 P(x)}$ Using the mean $\mu = 0.933$, let's calculate $\sigma$.

I'll calculate the standard deviation next.The mean $\mu$ is approximately $0.933$, and the standard deviation $\sigma$ is approximately $0.929$, both rounded to the nearest thousandth.

Summary:

1. Probability of 1 medical test: $\frac{1}{3}$
2. Probability of less than 2 tests: $\frac{11}{15}$
3. Probability of 0 to 1 tests (inclusive): $\frac{11}{15}$
4. Mean ( $\mu$ ): $0.933$
5. Standard Deviation ( $\sigma$ ): $0.929$

Would you like to dive deeper into any part of this? Here are some related questions:

1. What is the cumulative probability distribution for each $x$?
2. How is the variance different from the standard deviation?
3. How would this distribution change if one of the probabilities increased?
4. What is the probability of getting more than 1 test?
5. How can we visualize this probability distribution?

Tip: When working with probabilities, always ensure that the total probability adds up to 1!