A drug test is accurate 98% of the time.  If the test is given to 1700 people who have not taken drugs, what is the probability that at least 37 will test positive?

Probability = 

Give your answers to at least 3 decimal places.

We are dealing with a **binomial distribution** because the test has two possible outcomes for each person: either testing positive or not.

Given dataWe are dealing with a **binomial distribution** because the test has two possible outcomes for each person: either testing positive or not.

Given 

- The test is **98% accurate**, which means the probability of a false positive (incorrectly testing positive) is \( p = 1 - 0.98 = 0.02 \).
- The total number of people is \( n = 1700 \), and none of them have taken drugs (so we're concerned with the false positives).
- We are asked to find the probability that **at least 37 people** will test positive.

### Step 1: Approximate the binomial distribution
For large \( n \), the binomial distribution can be approximated by a **normal distribution** using the following parameters:

- Mean (\( \mu \)): 
  \[
  \mu = n \cdot p = 1700 \cdot 0.02 = 34
  \]
- Standard deviation (\( \sigma \)): 
  \[
  \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{1700 \cdot 0.02 \cdot 0.98} \approx 5.782
  \]

### Step 2: Use the normal approximation
We are now approximating the probability that at least 37 people test positive, i.e., \( P(X \geq 37) \).

Since we are using a normal approximation, we apply a **continuity correction**. The probability \( P(X \geq 37) \) becomes \( P(X \geq 36.5) \) in the normal distribution.

We standardize the normal variable using the **z-score** formula:
\[
z = \frac{x - \mu}{\sigma}
\]

For \( x = 36.5 \):
\[
z = \frac{36.5 - 34}{5.782} \approx 0.432
\]

### Step 3: Find the cumulative probability
Using a standard normal distribution table or calculator, we find the cumulative probability corresponding to \( z = 0.432 \).

\[
P(Z \leq 0.432) \approx 0.666
\]

Therefore, the probability that fewer than 37 people will test positive is approximately 0.666.

### Step 4: Find the final probability
The probability that **at least 37 people** will test positive is the complement of this value:

\[
P(X \geq 37) = 1 - 0.666 = 0.334
\]

### Final Answer:
The probability that at least 37 people will test positive is approximately **0.334** (to three decimal places).

---

Would you like more details, or do you have any questions? Here are five related questions you can consider:

1. What is the probability that exactly 37 people will test positive?
2. How would the result change if the test accuracy was 99%?
3. Can we solve this problem without the normal approximation?
4. How does the sample size (1700 people) impact the result?
5. What if the test accuracy was only 90%?

**Tip:** Always remember to apply the continuity correction when approximating a binomial distribution with a normal distribution.

A drug test is accurate 98% of the time. If the test is given to 1700 people who have not taken drugs, what is the probability that at least 37 will test positive?

Explore the probability that at least 37 out of 1700 people will test positive in a drug test with 98% accuracy, using binomial distribution and normal approximation. Learn the key formulas and steps involved in solving this probability problem. Suitable for undergraduate-level students.

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