The probabilities that a patient will have 0,1 ,2 , or 3 medical tests performed on entering a hospital are 615
, 5/15
, 3/15
, and 1/15
 respectively. Construct a probability distribution.

Medical Tests x
P(x)
 (Fractions only)
0	Answer 1 Question 3
6/15
1	Answer 2 Question 3
5/15
2	Answer 3 Question 3
3/15
3	Answer 4 Question 3
1/15
What is the probability of 1 medical test being performed? Simplify all fractions.

P(x=
 Answer 5 Question 3
1
 )=
 Answer 6 Question 3
1/3

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

P(x<
 Answer 7 Question 3
2
 )=
 Answer 8 Question 3
11/15

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

P(
 Answer 9 Question 3
0
 ≤x≤
 Answer 10 Question 3
2
 )=
 Answer 11 Question 3

What is the mean and standard deviation? Use decimals and round to the nearest thousandth.

μ=
 Answer 12 Question 3
0.933

σ=
 Answer 13 Question 3



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!

The probabilities that a patient will have 0, 1, 2, or 3 medical tests performed on entering a hospital are 6/15, 5/15, 3/15, and 1/15 respectively. Construct a probability distribution.

What is the probability of 1 medical test being performed? Simplify all fractions.

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

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

What is the mean and standard deviation? Use decimals and round to the nearest thousandth.

This problem focuses on constructing a probability distribution for the number of medical tests performed in a hospital, with probabilities 6/15, 5/15, 3/15, and 1/15 for 0 to 3 tests. It includes finding the probability of specific cases, such as 1 test, less than 2 tests, and between 0 to 1 tests, as well as calculating the mean and standard deviation. Suitable for students in Grades 10-12.

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