Half-Life & False Positive
Two problem types that look different but both follow a plug-and-play recipe. Learn the setup, and the rest is calculator work.
The Plug-and-Play Formula
Every half-life problem gives you three things and asks for the fourth. Identify what you have, plug in, and let the calculator finish.
The #1 setup trap: is always the starting amount — whatever they had at the “zero moment,” when the clock starts. is the target or leftover amount at some later time. If you swap them, the log flips sign and the answer breaks. Ask yourself: “which number was first?”
Worked Example — “When to give the second dose”
A patient is given 400 mg of medicine. The medicine has a half-life of 5.3 hours. The doctor wants the amount in the patient’s bloodstream to stay above 250 mg. When should the doctor give the second dose?
Sanity check: Half of 400 is 200. We’re trying to hit 250, which is more than half. So it shouldn’t take a full half-life (5.3 hours) to get there. 3.59 hours is less than 5.3 — that checks out.
Half-Life Practice (3)
Work each one on paper using the formula, then open the solution to check. Identify , , and HL before you touch the calculator.
1 · Antibiotic — when to give the next dose
A patient receives 600 mg of an antibiotic. The antibiotic has a half-life of 4 hours. The doctor wants the amount in the patient’s bloodstream to stay above 200 mg. When should the doctor administer the next dose?
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Sanity check: Half of 600 is 300. We need 200, which is below 300 — so it should take more than one half-life (4 hours). 6.34 hours > 4 hours. That tracks.
Common trap: Don’t put 200 in the slot. 200 is what’s left later, not what you started with. The dose came first — 600 is .
2 · Pain reliever — when does it drop to 75 mg?
A patient takes 500 mg of a pain reliever. The pain reliever has a half-life of 6.5 hours. When will the amount in the patient’s bloodstream fall to 75 mg?
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Sanity check: 75 is well below half of 500 (250), so this takes multiple half-lives. One HL = 6.5h → 250. Two HL = 13h → 125. Three HL = 19.5h → 62.5. We want 75, between 125 and 62.5 — so between 2 and 3 half-lives. 17.79 is between 13 and 19.5. Checks out.
Tip: Keep the fraction inside the log until you’re ready to hit the calculator. Writing or are both fine, but don’t evaluate and separately — rounding errors sneak in.
3 · Medication — when does it drop to 100 mg?
A patient is given 350 mg of a medication. The medication has a half-life of 3.8 hours. When will the amount in the patient’s bloodstream fall to 100 mg?
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Sanity check: Half of 350 is 175. We need 100, which is below 175 — more than one half-life. Two half-lives = 7.6h puts us at 87.5. We want 100, between 175 and 87.5 — so between 1 and 2 half-lives (3.8–7.6h). 6.87 fits right in that window.
The Population Table Method
These problems always give you three numbers: how accurate the test is, what percentage of people actually have the disease, and how many people are tested. You build a table, then the final answer is just one fraction.
The surprise every time: Even with a 90% accurate test, if the disease is rare (say 5% of people), most positive results are false positives. The not-infected group is so much bigger that even a small error rate produces a huge pile of wrong results. That’s why the final probability is way lower than you’d expect.
Worked Example — Disease Test
A disease test is 90% accurate. 5% of the population is infected with the disease. 10,000 people are tested.
Step 4 is the trap. “If the test is 90% accurate, what percentage of tests are wrong?” — 10%. That 10% error rate hits the not-infected group, which is 9,500 people. 10% of 9,500 = 950 false positives. That’s more than double the 450 true positives, even though the test is “90% accurate.”
False Positive Practice (3)
Same structure every time: split the population, apply the accuracy to each group, count total positives, then do the fraction. Work it on paper first.
1 · Disease screening — 85% accurate, 3% infected
A disease test is 85% accurate. 3% of the population is infected with the disease. 20,000 people are tested.
(a) How many people have the disease?
(b) How many people do not have the disease?
(c) How many people have the disease and test positive?
(d) How many people do not have the disease but test positive?
(e) If someone tests positive, what is the probability they actually have the disease?
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Notice: The test is 85% accurate, but only 14.9% of positive results are real. When the disease is rare (3%), the false positives from the massive healthy group swamp the true positives — 2,910 false alarms vs. 510 real cases.
2 · Workplace drug test — 95% accurate, 8% use
A workplace drug test is 95% accurate. 8% of employees actually use the substance being tested. 5,000 employees are tested.
(a) How many employees use the substance?
(b) How many employees do not?
(c) How many users test positive?
(d) How many non-users test positive?
(e) If an employee tests positive, what is the probability they actually use the substance?
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Notice: Higher accuracy (95%) and higher prevalence (8%) both help. Compare to problem 1: test 85%→95%, rate 3%→8%. Result: 62.3% vs. 14.9%. Two things drive the answer — test quality and how common the condition is.
3 · Allergy screening — 88% accurate, 10% allergic
An allergy test is 88% accurate. 10% of patients at a clinic are allergic to a specific substance. 5,000 patients are tested.
(a) How many patients are allergic?
(b) How many patients are not allergic?
(c) How many allergic patients test positive?
(d) How many non-allergic patients test positive?
(e) If a patient tests positive, what is the probability they are actually allergic?
Show solution ›
Notice: Even with 10% prevalence, the false positives (540) still outnumber the true positives (440). The non-allergic group has 4,500 people — 12% of that is 540, which beats 88% of 500 (440). The ratio flips only when prevalence gets high enough or accuracy nears 100%.
Before You Leave — Add This to Your Sheet
Half-life: . is always the starting amount. Sanity check: should the answer be more or less than one half-life?
False positive: Split the population → apply accuracy to each group → add true + false positives → fraction: . Rare diseases + imperfect tests = lots of false alarms.