A sample of 27 people yielded the information about their health insurance in the table to the right.
Two people who provided information for the table were selected at​ random, without replacement. Find the probability that neither had traditional insurance.
Number of People
 Type of Insurance
 
19
 Managed care plan
5
 Traditional insurance
3
 No insurance
 


To solve this problem, we need to find the probability that neither of the two people selected had traditional insurance. Here's how we can approach it:

### Step 1: Total Number of People
There are 27 people in the sample, as the table indicates:

- 19 have a managed care plan,
- 5 have traditional insurance,
- 3 have no insurance.

### Step 2: Total Number of Ways to Select 2 People
We are selecting 2 people without replacement from the total of 27 people. The total number of ways to select 2 people from 27 is given by the combination formula:

\[
\text{Total combinations} = \binom{27}{2} = \frac{27 \times 26}{2} = 351
\]

### Step 3: Number of People Without Traditional Insurance
The number of people without traditional insurance is the sum of those with a managed care plan (19 people) and those without insurance (3 people). So:

\[
19 + 3 = 22 \text{ people}
\]

### Step 4: Number of Ways to Select 2 People Without Traditional Insurance
We want to find the number of ways to select 2 people from these 22 people who do **not** have traditional insurance. The number of ways to do this is:

\[
\text{Combinations of 2 from 22} = \binom{22}{2} = \frac{22 \times 21}{2} = 231
\]

### Step 5: Probability Calculation
The probability that neither of the two selected people had traditional insurance is the ratio of favorable outcomes (selecting 2 people without traditional insurance) to the total number of outcomes (selecting 2 people from the entire group):

\[
P(\text{neither has traditional insurance}) = \frac{231}{351}
\]

Simplifying this fraction:

\[
P = \frac{231}{351} = \frac{22}{33} \approx 0.6667
\]

So, the probability is approximately **0.667** or **66.67%**.

Would you like further details on this or have any questions?

Here are 5 related questions to extend this:

1. What is the probability that both selected people have traditional insurance?
2. What is the probability that one person selected has traditional insurance and the other does not?
3. How would the probability change if we were to select 3 people instead of 2?
4. Can you explain how the combination formula works in this context?
5. What is the impact of sampling without replacement on this problem?

**Tip:** When calculating probabilities with combinations, always ensure you're clear on the size of the groups involved to avoid errors in counting outcomes.

A sample of 27 people yielded the information about their health insurance in the table to the right. Two people who provided information for the table were selected at​ random, without replacement. Find the probability that neither had traditional insurance.
Number of People
Type of Insurance
19 - Managed care plan
5 - Traditional insurance
3 - No insurance.

Learn how to calculate the probability that neither of two randomly selected individuals from a sample of 27 has traditional insurance. This problem uses combinations and basic probability principles, suitable for high school students. A detailed, step-by-step explanation is provided, covering managed care plans, traditional insurance, and those without insurance.

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