Calculator Inputs
Expansion Table
This table shows each binomial term from k = 0 to k = n.
| Trait count k | Coefficient | Expansion term | Probability | Percent |
|---|---|---|---|---|
| 0 | 1 | 1p^0q^4 | 0.003906 | 0.390625% |
| 1 | 4 | 4p^1q^3 | 0.046875 | 4.6875% |
| 2 | 6 | 6p^2q^2 | 0.210938 | 21.09375% |
| 3 | 4 | 4p^3q^1 | 0.421875 | 42.1875% |
| 4 | 1 | 1p^4q^0 | 0.316406 | 31.640625% |
Example Data Table
Example: four offspring with a dominant trait probability of 0.75.
| n | k | p | q | Formula | Probability |
|---|---|---|---|---|---|
| 4 | 0 | 0.75 | 0.25 | 1 × 0.750 × 0.254 | 0.00390625 |
| 4 | 1 | 0.75 | 0.25 | 4 × 0.751 × 0.253 | 0.046875 |
| 4 | 2 | 0.75 | 0.25 | 6 × 0.752 × 0.252 | 0.2109375 |
| 4 | 3 | 0.75 | 0.25 | 4 × 0.753 × 0.251 | 0.421875 |
| 4 | 4 | 0.75 | 0.25 | 1 × 0.754 × 0.250 | 0.31640625 |
Formula Used
The calculator uses the binomial expansion: (p + q)n. Here, p is the probability of the selected genetic outcome. The value q is the probability of the other outcome. Since there are only two outcomes, q equals 1 minus p.
The exact probability is: P(X = k) = nCk × pk × qn-k. The coefficient nCk counts how many arrangements can place k selected outcomes among n offspring. The calculator also sums exact terms for at least, at most, and between questions.
Mean is n × p. Variance is n × p × q. Standard deviation is the square root of variance.
How to Use This Calculator
Enter the total number of offspring or independent genetic trials. Add the target count for the trait you want to study. Enter the trait probability as a decimal. For example, use 0.75 for a three to one dominant phenotype ratio.
Choose the probability type. Use exact for one specific count. Use at least or at most for cumulative probability. Use between when your question has a lower and upper count. Add a sample size to estimate how many families or crosses may match the result.
Press calculate. The result appears below the header and above the form. Review the expansion table for every term. Use the CSV or PDF buttons to save the result.
About the Binomial Expansion Equation Genetics Calculator
Purpose
This calculator helps study genetic outcomes with binomial expansion. It is useful when each offspring has two possible result groups. One group may be a dominant phenotype. Another group may be a recessive phenotype. The same method also works for sex ratio questions. It can also model carrier probability in simple cases.
Why Binomial Expansion Fits Genetics
Many Mendelian probability questions repeat the same event. Each child or offspring is treated as one trial. The probability should stay constant for every trial. The outcome of one trial should not change another trial. When those assumptions are reasonable, binomial expansion gives fast answers. It lists every possible count pattern.
What the Result Means
The exact result shows one selected count. A cumulative result adds several exact terms. This is helpful for questions like at least two affected offspring. It also helps when the answer asks for no more than one carrier. Expected count converts probability into a planning estimate. It does not guarantee a real sample result.
Coefficient and Term Details
The coefficient counts arrangements. For example, two selected outcomes among four offspring can appear in six orders. Each order has the same probability when p and q are fixed. The calculator multiplies the coefficient by p powers and q powers. This creates the full probability term.
Practical Notes
Use decimals for probabilities. Convert percentages before entry. For 25 percent, enter 0.25. For a three to one ratio, enter 0.75 for the dominant group. Very large trial counts can create long tables. This page limits the value to keep results readable. Always match the probability to the selected trait label.
FAQs
What does p mean in this calculator?
p is the probability of the selected genetic outcome. It may represent a phenotype, genotype class, carrier state, or any chosen two-outcome trait group.
What does q mean?
q is the probability of the other outcome. The calculator sets q as 1 minus p because the model uses two outcome groups.
Can this calculator handle dominant phenotype ratios?
Yes. For a classic three to one dominant phenotype ratio, enter p as 0.75 for the dominant phenotype and use q as 0.25.
What is nCk?
nCk is the binomial coefficient. It counts how many ways k selected outcomes can be arranged among n total trials.
When should I use at least?
Use at least when the question asks for k or more selected offspring. The calculator adds all exact probabilities from k through n.
When should I use at most?
Use at most when the question asks for k or fewer selected offspring. The calculator adds all exact probabilities from 0 through k.
Can I use this for genotype probability?
Yes, when the genotype class has a fixed probability in every trial. Enter that genotype probability as p and define the other outcomes as q.
Why is the expected count not a guarantee?
Expected count is a long-run average estimate. Real families or crosses can differ because inheritance outcomes vary by chance.