Step 1: Understand the data

The table provides the following information:

• Total test subjects: $15 + 42 + 32 + 9 = 98$
• Number of false positive results: 15

We are tasked with finding:

1. The probability of selecting 2 subjects who both had false positive results (without replacement).
2. Whether it is unusual for this to happen.
3. An explanation for the determination of "unusual."

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Step 2: Probability calculation

The probability of selecting 2 false positive results without replacement is calculated as:

$P(\text{2 false positives}) = \frac{\text{Number of ways to choose 2 false positives}}{\text{Total number of ways to choose 2 subjects}}$

Numerator: Choosing 2 false positives

The number of ways to choose 2 subjects from the 15 false positives is calculated using the combination formula:

$\binom{15}{2} = \frac{15 \cdot 14}{2} = 105$

Denominator: Choosing 2 subjects from all 98

The total number of ways to choose 2 subjects from the 98 test subjects is:

$\binom{98}{2} = \frac{98 \cdot 97}{2} = 4753$

Probability

Now, calculate the probability:

P(\text{2 false positives}) = \frac{105}{4753} \approx 0.0221 \, \text{(or 2.21%)}

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Step 3: Determine if it is unusual

An event is generally considered unusual if its probability is less than 5% ($P < 0.05$).

• The calculated probability ($0.0221$) is less than 5%, so it is unusual to randomly select 2 subjects and get 2 false positive results.

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Final Answers:

1. Probability: $0.0221$ (or $2.21\%$).
2. Is it unusual?: Yes.
3. Why?: The probability is very low ($< 5\%$).

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Would you like a breakdown of any steps or further clarification? 😊

Related Questions:

1. What is the probability of selecting one true positive and one true negative?
2. What is the probability of selecting one false negative and one true negative?
3. How do combinations differ from permutations, and when are they used?
4. If replacement was allowed, how would the probability change?
5. How does the concept of "unusual" apply in other statistical contexts?

Tip:

For "without replacement" scenarios, always ensure that the total number of available items decreases for subsequent selections. 🎯