Chance sampling changes frequencies even when variants have equal expected reproductive success.
By the end: Explain drift, loss, fixation, bottlenecks, and the difference between census and effective population size.
Sampling adds variation around an expectation
Even without a fitness difference, only a finite set of gene copies reaches the next generation. Some parental copies leave more descendants by chance. Genetic drift is the resulting stochastic change in allele frequencies. Neutrality means equal expected contribution, not identical realized offspring counts.
The lab uses a haploid Wright-Fisher model: each new generation has N gene copies, sampled independently from the previous frequency p. Its A count is binomial. Expected next frequency remains p, but its variance is p times one minus p divided by N. Smaller populations have a wider spread of possible next frequencies. For four copies at p = 0.25, exactly one A can occupy four different positions. Each order has probability 0.25 × 0.75³, so the total probability is four times that, about 42.19%.
Count the possible reproductive samples
Imagine four haploid parents, one carrying A and three carrying a. Haploid means each parent has one copy at the locus, so both the organism count and the gene-copy count are four here. The next generation also contains four copies. Each new copy independently chooses a parental copy with equal probability. A parent can supply several descendants, or none. Sampling with replacement describes this inheritance model; it does not mean physically removing and returning DNA from a parent.
The model distinguishes a parental identity from an allele label. Suppose all four offspring happen to copy the same a parent. That parent's particular copy has four descendants, while the other three parental copies have none. The resulting A count is zero. Another possible sample includes the A parent twice and two different a parents once each. That produces two A copies. Equal probabilities for parental contributions do not require equal realized contributions in either sample.
Parental identities and two possible samples
Constructed inheritance samples, not measured families. Circles carry A; squares carry a. Subscripts here are identity numbers written as ordinary digits. Paths show repeated copying, not physical movement. Geometry is schematic; the two illustrated samples are not equally weighted categories.
For a particular ordered sample containing one A followed by three a copies, multiply the four independent probabilities. The result is one quarter times three quarters times three quarters times three quarters, or 27/256. But this is only one ordering. The A copy could instead occupy the second, third, or fourth position. Those four mutually exclusive orders have equal probabilities, so exactly one A has probability 108/256. Counting orders prevents confusing one inheritance path with the complete outcome.
The remaining counts follow the same reasoning. Exactly two A copies can occupy six pairs of positions, giving 54/256. Exactly three A copies have four possible positions for the lone a, giving 12/256. Four A copies have probability 1/256. Zero A copies have probability 81/256. Adding the five numerators gives 256, so the probabilities sum to one. These are constructed model probabilities, not observed counts from a biological population or five equally likely futures.
Now weight each possible count by its probability. Zero times 81, plus one times 108, plus two times 54, plus three times 12, plus four times one equals 256. Dividing by 256 gives one expected A copy. Dividing again by four total copies gives an expected frequency of one quarter. This average includes populations that lose A completely and populations in which A increases. An expected count of one is compatible with zero being a common outcome.
All four-copy outcomes
Next A count
Next A frequency
Probability
0
0%
81/256
1
25%
108/256
2
50%
54/256
3
75%
12/256
4
100%
1/256
Exact constructed neutral distribution. Probabilities sum to one; the probability-weighted mean count is one. Rows are alternatives, not successive generations.
Drift therefore describes sampling variation around an expectation, rather than a hidden advantage assigned to the eventual winner. In this example the expected A frequency does not move, yet the next population need not resemble its parents. A real observation of zero A in a carefully counted descendant population would be consistent with this neutral mechanism, but would not prove neutrality. Selection or a biased population sample might also produce that observation. The calculation tests what chance can produce under specified assumptions, not which assumptions nature necessarily satisfies.
Loss and fixation are boundary states
Starting with A frequency one quarter and four gene copies, the chance of losing A in the next generation is three quarters to the fourth power, about 31.64%. With twenty copies at the same frequency, it is about 0.317%. Equal expected frequency does not imply a small chance of disappearance.
Once A is lost, this model cannot restore it; once A is fixed, only A remains. In a finite neutral population with no mutation or migration and continued reproduction under this model, eventual fixation probability equals the starting frequency. That is not the chance of fixation next generation and is not a prediction with selection or changing conditions.
Separate immediate loss from eventual fixation
A boundary state has a special property in the closed model. If the current A frequency is zero, every possible parental choice supplies a. The next frequency must also be zero. If the current frequency is one, every parental choice supplies A. Selection among the remaining copies cannot recreate the absent alternative. Mutation or genetic input could change that conclusion, but those are additional mechanisms. A boundary is permanent only relative to the mechanisms excluded from the model.
Starting with one A among four copies, fixation in the very next generation requires four A draws. Its probability is one quarter raised to the fourth power, or 1/256, about 0.39%. Loss next generation instead requires four a draws and has probability 81/256, about 31.64%. Neither number is the eventual fixation probability of 25%. The first two concern one reproductive step; the last allows any number of intervening generations before a boundary is reached.
One way to understand eventual fixation is to follow many independent populations with the same initial frequency. Their average frequency stays one quarter at each generation under neutrality. Over sufficiently long histories, a finite closed Wright-Fisher population reaches loss or fixation with probability one. At that point each population contributes either zero or one to the frequency average. An average of one quarter then means one quarter of the populations fixed A. This argument concerns repeated hypothetical populations, not a guarantee for one population.
Three different boundary questions
Question from one A among four
Probability
Time horizon
Loss
81/256 ≈ 31.64%
Next generation
Fixation
1/256 ≈ 0.39%
Next generation
Fixation under continued neutral sampling
25%
Eventually
Constructed closed finite Wright-Fisher model. Eventual fixation assumes no mutation, migration, or selection; the two immediate outcomes do not exhaust all possible next states.
Increasing population size changes the size of sampling fluctuations without changing their neutral direction. At an A frequency of one quarter, four copies have frequency variance 0.25 times 0.75 divided by four, or 0.046875. Twenty copies give 0.009375, one fifth as much. Variance averages squared deviations from the expected frequency. Taking square roots gives standard deviations of about 0.2165 and 0.0968 in frequency units. Those are spreads of possible frequencies, not chances of losing A.
There is another comparison that gives a different answer. One A among four copies starts at 25%, whereas one A among twenty starts at 5%. The larger population in that comparison has a lower starting frequency and a lower eventual neutral fixation probability. The lab avoids this change of question by keeping frequency fixed at 25%, so twenty copies begin with five A copies. Always ask whether a comparison holds the allele's count fixed or holds its frequency fixed.
Hold frequency or hold count?
Comparison
Four copies
Twenty copies
Hold starting frequency at 25%
1 A
5 A
Eventual neutral fixation
25%
25%
Hold starting A count at one
25% A
5% A
Eventual neutral fixation
25%
5%
Constructed neutral cases. Population size alone does not specify the initial frequency or eventual fixation probability.
Do not translate these neutral boundary calculations into claims about adaptive merit. A fixed allele could have reached fixation through drift without improving any trait. A favorable allele can disappear while rare because its higher expected contribution still permits a sample containing no descendants. Environmental conditions determine any fitness difference, and finite reproduction determines which contributions actually occur. Explaining a particular history requires both parts when both matter. Fixation describes which allele remains, not why it remained or whether it is useful.
Work through the mechanism
1. Chance that every sampled copy is a
N is the number of independently sampled haploid copies, not twice the number of individuals in this model.
2. Spread of next-generation frequency
The mean frequency stays p. Variance is the mean squared deviation around it, not the probability of loss.
Compare one generation of neutral sampling
Chance of losing A next generation: reference 31.64%; selected 3.17%. Expected frequency is 25% in both.
Selected count probabilities
A copies
A frequency
Probability
0
0%
3.17%
1
8.33%
12.67%
2
16.67%
23.23%
3
25%
25.81%
4
33.33%
19.36%
5
41.67%
10.32%
6
50%
4.01%
7
58.33%
1.15%
8
66.67%
0.24%
9
75%
0.04%
10
83.33%
<0.01%
11
91.67%
<0.01%
12
100%
<0.01%
Each generation samples N copies independently at p = 0.25. Bar height is probability mass; the gold bar at frequency zero is loss. Both axes match. Multiples of four keep the initial A count integral. This enumerates all possible next counts, not a random animation. Controls replace the selected population size immediately; the four-copy reference stays visible. Displayed probabilities are rounded; very small positive values are not reported as zero.
Gene copies, p = 0.25
Chance A is lost next generation
4
0.75⁴ ≈ 31.64%
12
0.75¹² ≈ 3.17%
20
0.75²⁰ ≈ 0.317%
The lab enumerates the entire one-generation distribution; it does not show a chosen lucky trajectory.
Read probability mass rather than a chosen trajectory
Begin the drift lab with the severe-bottleneck preset. Both panels then represent four haploid gene copies, starting at an A frequency of 25%. Their identical distributions are a check that the comparison uses the same rule on both sides. The horizontal position is next-generation A frequency. The vertical height is the probability of that exact outcome. The leftmost bar is not an initial time point. It is the possible outcome of losing A at the next reproductive step.
Before choosing twenty copies, predict which feature will stay unchanged. The expected frequency remains 25%, while probability becomes more concentrated around it. With four copies, the only possible frequencies are zero, one quarter, one half, three quarters, and one. With twenty copies, adjacent outcomes differ by one twentieth, or five percentage points. More possible positions do not imply more uncertainty. Their individual probabilities combine into a distribution that is narrower in frequency even though it contains more bars.
Possible frequencies become more closely spaced
Constructed possible outcomes, not probabilities. Horizontal positions share a scaled zero-to-one frequency axis. Marker height and size are schematic and do not encode probability mass.
Open the selected count probabilities to connect the graph to exact events. At twenty copies, loss has probability three quarters raised to the twentieth power, about 0.003171 as a fraction, or 0.3171%. Multiplying a probability by one hundred converts it to a percentage. Confusing 0.003171 with 0.003171% makes the event appear one hundred times rarer. A positive value displayed below a rounding threshold is still possible. A bar too short to notice is not evidence that its outcome is forbidden.
A useful paper comparison asks about count spread instead of frequency spread. The expected A count is one for four copies and five for twenty copies. Count variance is N times p times one minus p, giving 0.75 and 3.75 respectively. Thus the larger population has a wider spread measured in numbers of A copies but a narrower spread measured as fractions of all copies. The two statements agree because dividing the count by a larger denominator changes the scale.
Count spread and frequency spread
Quantity at p = 0.25
N = 4 copies
N = 20 copies
Expected A count
1
5
A count variance
0.75 copies²
3.75 copies²
Expected A frequency
0.25
0.25
A frequency variance
0.046875
0.009375
Constructed binomial samples. Frequency is count divided by N, so frequency variance is count variance divided by N². Wider count spread can coexist with narrower frequency spread.
Compare the intermediate preset with the two endpoints. Twelve copies give a loss probability of three quarters raised to the twelfth power, about 3.1676%. That lies between the four-copy and twenty-copy risks, but not halfway between them. Requiring every draw to miss A multiplies the failure probabilities repeatedly. The relationship is exponential rather than a linear decrease per added copy. The preset changes the next-generation distribution immediately; it does not replay a population shrinking or growing through those sizes.
Finally ask a question this widget cannot answer directly. What fraction of populations have lost A after ten generations? The displayed distribution supplies only the first step. After that step, different populations have different frequencies and therefore different probabilities for subsequent samples. Reusing the initial loss probability ten times would ignore those changes. A multigeneration calculation must carry forward every possible count and its probability, including the absorbing boundaries. The static comparison isolates one sampling mechanism rather than simulating a particular lucky lineage.
Population history matters
A bottleneck sharply reduces a population; a founder event starts a new population from a subset. Either can change frequencies and reduce variation through sampling, even without an adaptive advantage. Recovery of census size does not automatically recreate lost alleles.
Effective population size describes the strength of drift under a specified genetic model. Unequal reproductive success, unbalanced breeding sex ratios, and changing population size can make it much smaller than a head count. Real populations also combine drift with selection. A useful allele can be lost while rare; not every fixed difference is evidence of adaptation.
Connect sampling to demographic history
A bottleneck reduces the pool that can contribute descendants. A founder event establishes a population using a subset of a source population. Either can produce allele frequencies unlike the earlier population without a fitness difference. Consider a constructed haploid source containing ten A and ninety a copies. Four founders happen to include two of each type. Their A frequency is one half, compared with one tenth in the source. This identifies a changed starting composition, not evidence that A improved survival or reproduction.
Suppose those founders then produce a population of four hundred descendants, with equal contribution from each founder and no mutation or immigration. The two A founders supply two hundred A copies. The frequency remains one half even though census size increases one hundredfold. If a third allele was absent among the founders, multiplying their descendants cannot restore it. Demographic recovery and recovery of lost variation are separate processes. A large present population can therefore retain genetic consequences of a much smaller ancestral population.
Growth repeats the founders' copies
Constructed founder history. Each lower symbol stands for 100 descendants of the founder above it; upper symbols each stand for one founder. No mutation, immigration, or unequal contribution. Geometry and population areas are schematic.
Sampling survivors without replacement is different from the offspring model used in the lab. In a disaster that leaves exactly four of one hundred existing individuals alive, no individual can appear twice among the survivors. The lab instead allows the same parental copy to supply several offspring. Both can change frequencies through chance, but their exact probability formulas differ. Do not apply the binomial loss formula to every bottleneck merely because the surviving head count is four. Specify the sampling process first.
Nor is every disaster genetically random. If a heritable trait changes survival during the event, selection changes which copies are likely to remain. A flood could also remove one geographic patch more than another, changing frequencies through spatial structure. Calling an event a bottleneck describes the reduction in numbers; it does not prove equal survival probabilities. Evidence about exposure, traits, and ancestry helps distinguish random sampling from systematic differences. Several mechanisms can contribute to the same reduction in population size.
Effective population size asks which idealized population would show a comparable strength of drift under a stated definition. A census of one hundred adults may include many that contribute no offspring. If only a few families supply most descendants, those descendants share more recent ancestral copies than equal parental contribution would suggest. Drift can then be stronger than the head count alone predicts. Effective size is not obtained simply by counting adults or automatically counting every breeder as an independent, equally contributing parent.
A reduction in numbers does not identify its cause
Candidate mechanism
Relevant comparison
What remains uncertain
Neutral survival sampling
Survival unrelated to focal inherited type
Chance can still change frequency
Selection during the event
Survival differs by inherited type under exposure
Other life stages may offset this effect
Spatially uneven survival
Exposure differs across patches with different frequencies
Direct allele effects are not established
Hypothesis comparison, not empirical results. A bottleneck describes a population reduction; it does not establish random survival.
For an unfamiliar case, separate the time points before proposing an adaptive explanation. First identify which copies existed before the bottleneck. Next ask which entered the surviving or founding pool, and whether entry depended on the focal allele. Then distinguish later population growth from later genetic input. A frequent allele in the recovered population may record inherited sampling history rather than selection for a present function. Testing that history requires population and ancestry evidence, not a story based only on what the allele might do today.
Exercises
Expected frequency versus loss
For four neutral haploid copies starting with one A, what is expected A count next generation? What is the chance of zero A?
Hint
An expectation averages over all possible counts.
Worked answer
Outcome or summary
Value
Expected A count
1
Probability of zero A
81/256 ≈ 31.64%
Eventual neutral fixation probability
25%
The last probability refers to a different, multi-generation question.
Expected count is 4 × 0.25 = 1. The chance of zero is 0.75⁴ = 81/256, about 31.64%. Other outcomes, including counts above one, balance the expectation. Expected count one does not guarantee that even one copy survives.
Can a beneficial variant disappear?
Start with one A among four haploid copies. Give A relative reproductive contribution two and a contribution one. First form the weighted reproductive pool, then sample four offspring copies independently from it. Find the chance that no A reaches the next generation.
Hint
A contributes weight two and the three a copies contribute total weight three. Normalize before calculating the chance of four a draws.
Worked answer
Reproductive model
A probability per offspring
Chance of losing A
Equal contributions
0.25
31.64%
A contributes twice as much
0.40
12.96%
Both outcomes use four independent offspring draws. A reproductive advantage reduces, but does not eliminate, the chance of loss.
The reproductive pool has A frequency 2/(2 + 3) = 0.4. Each offspring draw therefore has probability 0.6 of being a, so complete loss of A has probability 0.6⁴ = 0.1296, or 12.96%. This is lower than the neutral 31.64%, but it is not zero. Selection changes the sampling probabilities; finite reproduction can still lose a beneficial variant. This combines two stated model steps rather than assuming selection and drift are mutually exclusive.
Sources and further reading
The explanations and exercises are complete without these optional links. The references cover inheritance, population mechanisms, and evidence for ancestry. Source annotations distinguish textbook explanations, secondary history, and a primary paper whose bibliographic identity alone was checked. The examples are constructed teaching models, not empirical estimates or professional decision advice.
Genetic variation, allele frequencies, and the Hardy-Weinberg reference model. Its closing claim about any genotype deviation implying evolution is too broad: sampling, subdivision, and nonrandom pairing can produce departures without an allele-frequency change. Our lessons distinguish the random-union calculation from stability across generations.