Study guide: Exponential decay, uncertainty, and model inference
Apply half-life models, interpret probabilistic counts, and infer a rate from bounded evidence.
Ideas to explain in your own words
- Apply an exponential model while distinguishing an expected amount from a tiny-sample guarantee.
- Separate fractional decay from a fixed subtraction.
- Distinguish an expected surviving count from a guaranteed integer count.
- Use ratios and intervals to assess an exponential-model explanation.
Watch your reasoning
- 12 hours contains four 3-hour half-lives: 80 → 40 → 20 → 10 → 5. Subtracting 12 units would assume a fixed hourly loss, which is not the supplied exponential model.
- 12/4 = 3 half-lives, so 240×(1/2)³ = 30 units.
- 3×1/2 = 1.5 is a mean over repeated samples. Each actual remaining count is an integer, so the expectation is not a literal partly decayed nucleus.
- The measured decrease has two possible contributors. The decay-only inference requires the stated no-addition and no-removal assumptions.
A worked review example
Calculate a remaining amount after half-lives: At 0, 3, 6 and 9 hours, the model gives 80, 40, 20 and 10 units. At 6 hours, t/3 = 2, so 80 × (1/2)² = 20. Calculate a Remaining Fraction: At 6 h, fraction remaining is 1/8 = 12.5%. Amount remaining is 160/8 = 20 parent units; 160 − 20 = 140 parent units transformed. Explain Tiny-Sample Decay Variation: Expected count is 2. All four remaining has probability 1/16 = 6.25%. A particular sample could contain three remaining; expectation averages outcomes across repetitions of the model. Infer a Half-life and Check a Claim: Two consecutive five-hour ratios equal 1/2, fitting h = 5 h and predicting 96×(1/2)³ = 12 units at 15 h. With unknown leakage, the same observed decline could combine decay and removal in different proportions.
Make a practice plan
- Try the supplied lesson problems without looking at the answers.
- Compare your method with the worked examples and explain any correction.
- Use the new review cases, then complete the transfer task on paper or in your own document.