About 33%, or roughly 1 in 3

This assumes "99% accurate" means the test correctly flags 99% of users (sensitivity) and correctly clears 99% of non-users (specificity).

The calculation (Bayes' theorem)

$$ P(\text{user} \mid +) = \frac{P(+ \mid \text{user})\,P(\text{user})}{P(+ \mid \text{user})\,P(\text{user}) + P(+ \mid \text{non-user})\,P(\text{non-user})} $$

$$ = \frac{0.99 \times 0.005}{0.99 \times 0.005 + 0.01 \times 0.995} = \frac{0.00495}{0.0149} \approx 0.332 $$

The intuitive version: imagine 10,000 people

Of about 150 positive results, only about 50 are real users. The other ~100 are false positives. So a positive result means roughly a 1 in 3 chance of actual drug use.

Why it surprises people

1. Base rate neglect. People focus on the vivid "99% accurate" figure and ignore how rare drug use is. When a condition is rare, even a small error rate applied to the huge non-user population produces more false positives than there are true positives.

2. Confusing the two conditional probabilities. The test tells you P(positive | user) = 99%. People instinctively treat that as P(user | positive). These are very different numbers. "Most users test positive" does not mean "most positives are users."

3. The 1% error feels negligible. But 1% of 9,950 non-users (~100 people) is double the entire user population (50 people).

Practical takeaway

This is why positive screening results are confirmed with a second, independent test. If someone tests positive twice, the probability jumps to about 98%, because the first positive raises the "base rate" from 0.5% to 33% before the second test is applied. The same logic applies to medical screening for rare diseases, fraud detection, and security screening.

The probability that the person actually uses the drug is 33.2% (or roughly 1 in 3).

This means that even with a 99% accurate test, a positive result is twice as likely to be a false alarm as it is to be a true positive.

Here is the breakdown of the math, followed by why our brains find this so hard to accept.

The Math (Made Simple)

While this is a classic application of Bayes' Theorem, it is much easier to understand using concrete numbers. Imagine we test a random population of 10,000 people.

1. Separate the users from the non-users:

• Users: 0.5% of 10,000 = 50 people actually use the drug.

• Non-Users: 99.5% of 10,000 = 9,950 people do not use the drug.

2. Test the 50 Users (True Positives):

• The test is 99% accurate, so it catches 99% of the 50 users.

• 0.99 × 50 = 49.5 people test positive.

3. Test the 9,950 Non-Users (False Positives):

• The test is 99% accurate, meaning it has a 1% failure rate. It will falsely accuse 1% of the innocent people.

• 0.01 × 9,950 = 99.5 people test positive.

4. Calculate the final probability:

• In total, 149 people got a positive test result (49.5 true + 99.5 false).

• Out of those 149 people, only 49.5 are actual drug users.

• 49.5 divided by 149 = 33.2%.

(Note: This assumes "99% accurate" means the test is 99% sensitive at catching users AND 99% specific at clearing non-users).

Why Does This Surprise People?

Most people intuitively guess the answer is 99%, or perhaps 98.5%. The real answer shocks people because of a well-documented psychological blind spot known as the Base Rate Fallacy (or Base Rate Neglect).

Here is why our intuition fails us:

1. We ignore the size of the pools When the human brain hears "99% accurate," it latches onto that number because it sounds like a near-certainty. Simultaneously, we ignore the "base rate" (the 0.5% of users). We forget that the 1% error rate is being applied to a massive group of non-users, while the 99% accuracy rate is being applied to a tiny group of actual users. 1% of a giant number is bigger than 99% of a tiny number.

2. We confuse the conditions People naturally confuse two very different probabilities:

• The probability of testing positive if you use the drug (99%).

• The probability of using the drug if you test positive (33.2%).

3. We are bad at dealing with percentages Evolutionary psychologists suggest that humans didn't evolve to understand percentages and probabilities; we evolved to understand tangible counts (like 3 out of 10 hunters). When the problem is framed as percentages (99%, 0.5%), almost everyone gets it wrong—including doctors and statisticians. When it is framed as frequencies (like the 10,000-person example above), the illusion disappears.

The Real-World Takeaway

This is why medical professionals and employers rarely rely on a single screening test for rare conditions or infrequent drug use. If someone tests positive on an initial screening, they are given a second, different test.

If that same person takes a second independent test and tests positive again, the "base rate" has now shifted from 0.5% to 33.2%, and the probability that they are a user skyrockets to 98%.

The answer: about 33%

Even though the test is "99% accurate," a person who tests positive has only roughly a 1-in-3 chance of actually using the drug.

The calculation (Bayes' theorem)

Assume "99% accurate" means the test correctly identifies 99% of users (sensitivity) and correctly clears 99% of non-users (specificity).

• P(user) = 0.005, P(non-user) = 0.995

• P(positive | user) = 0.99

• P(positive | non-user) = 0.01

$$ P(\text{user} \mid +) = \frac{0.99 \times 0.005}{0.99 \times 0.005 + 0.01 \times 0.995} = \frac{0.00495}{0.00495 + 0.00995} \approx 0.332 $$

The intuitive version: imagine 10,000 people

Of the ~150 people who test positive, only ~50 actually use the drug. So 50 / 150 ≈ 33%.

Why it surprises people

1. Base-rate neglect. People focus on the test's 99% accuracy and ignore how rare drug use is. Because non-users outnumber users 199-to-1, even a tiny 1% false-positive rate applied to that huge group produces more false positives than the test produces true positives from the small group of users.

2. Confusing two different conditional probabilities. "99% accurate" tells you P(positive | user) — if you use, you'll probably test positive. People intuitively flip this into P(user | positive) — if you test positive, you probably use. These are not the same thing, and the gap between them is entirely driven by the base rate. (This flip is sometimes called the "inverse fallacy" or, in courtrooms, the "prosecutor's fallacy.")

3. "99%" sounds like certainty. It feels like a 1% error rate should mean 1% of results are wrong. But the relevant question isn't "how often is the test wrong?" — it's "among the positive results, how many are wrong?" And that depends on what you're screening for.

The practical takeaway

This is why screening for rare conditions is almost always followed by a second, confirmatory test. A second independent 99%-accurate test on the ~150 positives would raise the probability of true use from 33% to about 98%, because the prior is no longer 0.5% — it's 33%.

Answer: about 33%, not 99%.

Assumptions

"99% accurate" is interpreted as both:

• Sensitivity of 99%: 99% of users test positive

• Specificity of 99%: 99% of non-users test negative (a 1% false-positive rate)

Calculation (Bayes' theorem)

Imagine 100,000 people:

Total positives = 495 + 995 = 1,490

$$P(\text{user} \mid +) = \frac{495}{1{,}490} \approx 0.332$$

Or in formula form:

$$P(U|+) = \frac{0.99 \times 0.005}{0.99 \times 0.005 + 0.01 \times 0.995} \approx 33.2\%$$

Why it surprises people

1. Base rate neglect. People focus on the test's accuracy and ignore how rare the drug use is. Because users are so scarce (0.5%), even a small false-positive rate applied to the huge non-user population produces more false positives than the true positives from the small user group.

2. Confusing two different conditional probabilities.

• P(positive | user) = 99%, which is what the test's accuracy tells you

• P(user | positive) ≈ 33%, which is what you actually want to know

These are not the same, and intuition tends to treat them as if they were.

3. Percentages hide the counts. A 1% error rate sounds tiny, but 1% of 99,500 people is 995, which is about double the 495 true users who test positive.

Key takeaways

• The meaning of a positive result depends heavily on who is being tested. If the test were given to a group where 50% use the drug, a positive would be right about 99% of the time.

• This is why screening for rare conditions (diseases, drugs, terrorism, etc.) typically requires confirmatory testing.

• A helpful mental trick is to use natural frequencies ("495 out of 1,490") instead of probabilities. People reason about these far more accurately.

About 33.2%—assuming “99% accurate” means both 99% sensitivity and 99% specificity. That is, the test detects 99% of users and correctly gives a negative result to 99% of non-users.

Imagine testing 20,000 people:

Of the 298 positive results, only 99 are from actual users:

[ P(\text{uses drug}\mid\text{positive})=\frac{99}{99+199}\approx \boxed{33.2\%}. ]

Why it surprises people: We tend to confuse “the test is positive in 99% of users” with “99% of people who test positive are users.” Those are different probabilities. Because non-users vastly outnumber users, even a 1% false-positive rate produces more false positives than true positives. Ignoring this imbalance is called base-rate neglect.

If “99% accurate” refers only to overall accuracy, rather than sensitivity and specificity separately, there isn’t enough information to calculate the answer.