Explain Tiny-Sample Decay Variation
Distinguish an expected surviving count from a guaranteed integer count.
Supplied materials
- Use the supplied fictional observations or model. Paper and pencil are optional; spoken answers are welcome. No equipment or real experiment is needed.
Learn the idea
Individual radioactive decays are probabilistic. A smooth amount curve is useful for a large sample or an expected count; a tiny sample does not have to lose exactly half its nuclei in each interval. For this explicit paper model, each of four parent nuclei independently has probability 1/2 of still being parent after one half-life. No parent nuclei enter, and no other process removes them.
The expected remaining count is 4×1/2 = 2. Possible integer remaining counts are 0, 1, 2, 3, or 4. “All four remain” requires four survival outcomes, giving (1/2)⁴ = 1/16. “None remain” likewise has probability 1/16. Thus the two-nucleus expectation is not a required result in every four-nucleus sample, and an expectation can be fractional for other tiny initial counts.
- Calculate expected remaining count after one half-life.
- List the five possible integer remaining counts.
- Multiply four 1/2 probabilities to obtain the all-survive chance.
- Explain why a supplied sample with three remaining would not contradict an expectation of two.
See the method in action
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.
Check your reasoning
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See the answer and explanation
1.5 as an expectation, not a literal half-nucleus
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.
Take it further
Change to two elapsed half-lives with three initial independent nuclei. Each survives as parent with probability 1/4, so expected remaining count is 3/4 = 0.75 and the all-three-survive probability is (1/4)³ = 1/64.
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