A drug test is accurate 98% of the time.  If the test is given to 2400 people who have not taken drugs, what is the probability that at most 51 will test positive?

Probability = 

Give your answers to at least 3 decimal places.

We are given a drug test that is 98% accurate, meaning there is a 2% chance of a false positive (testing positive when the person has not taken drugs). We are asked to find the probability that at most 51 people will test positive out of 2400 people who have not taken drugs.

This is a **binomial probability problem** since each test can either result in a false positive or not.

### Step 1: Define the parameters
- \( n = 2400 \) (total number of people tested)
- \( p = 0.02 \) (probability of a false positive)
- \( 1 - p = 0.98 \) (probability of a correct negative)
- We are looking for the probability that at most 51 people test positive, i.e., \( P(X \leq 51) \), where \( X \) is the number of false positives and follows a binomial distribution \( X \sim \text{Bin}(n = 2400, p = 0.02) \).

### Step 2: Approximation with normal distribution
Since \( n \) is large and \( p \) is small, we can use the normal approximation to the binomial distribution. The mean \( \mu \) and variance \( \sigma^2 \) of the binomial distribution are given by:
- \( \mu = np = 2400 \times 0.02 = 48 \)
- \( \sigma^2 = np(1 - p) = 2400 \times 0.02 \times 0.98 = 47.04 \)
- \( \sigma = \sqrt{47.04} \approx 6.86 \)

Using the normal approximation, \( X \) can be approximated by a normal distribution \( N(\mu = 48, \sigma = 6.86) \).

### Step 3: Continuity correction and standardization
To improve the approximation, we apply the continuity correction. To find \( P(X \leq 51) \), we calculate \( P(X \leq 51.5) \).

We standardize the value:
\[
Z = \frac{X - \mu}{\sigma} = \frac{51.5 - 48}{6.86} \approx 0.51
\]

### Step 4: Find the probability
Using the standard normal distribution table, the cumulative probability corresponding to \( Z = 0.51 \) is approximately 0.6950.

Thus, the probability that at most 51 people will test positive is approximately:
\[
P(X \leq 51) \approx 0.695
\]

### Final Answer:
The probability that at most 51 people will test positive is approximately \( \boxed{0.695} \).

Do you want more details on the normal approximation process or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the probability that exactly 51 people will test positive?
2. How would the result change if the test were 99% accurate instead of 98%?
3. What is the expected number of people to test positive out of 2400?
4. How does the normal approximation differ from using the exact binomial distribution?
5. If 60 people test positive, what is the probability that this result is due to chance?

**Tip:** Normal approximation is useful when the sample size is large and the probability is not too close to 0 or 1.

A drug test is accurate 98% of the time. If the test is given to 2400 people who have not taken drugs, what is the probability that at most 51 will test positive? Probability = Give your answers to at least 3 decimal places.

Learn how to calculate the probability that at most 51 people test positive out of 2400 in a 98% accurate drug test. This problem applies binomial probability, normal approximation, and uses the Z-score formula for approximation. Suitable for undergraduate statistics students.

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