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A conditional probability distribution describes how one random variable behaves once you know the value of another. If Y is an outcome and X is something you have observed, the conditional distribution of Y given X = x is the full set of probabilities (or the density) for Y in the subgroup where X equals x. This guide gives the formulas for the discrete and continuous cases, one worked example of each with the arithmetic shown, reading guidance, reporting wording, and code for Excel, R and Python.
Definition in one paragraph
A conditional distribution is the distribution of a random variable, computed with the rules of conditional probability, after observing the value taken by another random variable (StatLect, “Conditional probability distributions”). The rule is “conditional = joint / marginal”: take the slice of the joint distribution at X = x, then divide by the marginal probability (or density) of X at x so the slice adds up to 1 (Davis Ross, STAT 350 handout 23). Because there is a separate answer for every value of x, you get a family of distributions, not a single one.
What is probability, and how do you find it?
Probability is a number from 0 to 1 that measures how likely an event is. For a discrete outcome you find it by adding the probabilities of the outcomes that make up the event; for a continuous outcome you find it from the area under a density curve. Conditioning changes the starting point: you ask for the probability of an event inside the group where something else is already known. The single-event version is covered in how to calculate conditional probability and conditional probability and Bayes’ theorem; this page extends it to whole distributions. The joint distribution you start from is explained in the joint probability formula guide.
Formula, with every symbol explained
Discrete case: the conditional probability mass function
pY|X(y | x) = P(Y = y | X = x) = pX,Y(x, y) / pX(x), for a fixed x with pX(x) > 0.
- pY|X(y | x): the conditional probability mass function of Y given X = x.
- pX,Y(x, y): the joint probability that X = x and Y = y at the same time.
- pX(x): the marginal probability that X = x, found by summing the joint probabilities over all values of y.
- x is held constant; y is the variable that runs over the outcomes. For background on mass functions, see the probability mass function guide.
Continuous case: what is a probability density function, and what is the conditional one?
A probability density function (pdf) describes a continuous variable: probabilities come from areas under the curve, and the height of the curve at a point is not itself a probability. The conditional pdf has the same “joint over marginal” shape:
fY|X(y | x) = fX,Y(x, y) / fX(x), for a fixed x with fX(x) > 0.
Here fX,Y is the joint density and fX(x) is the marginal density of X, found by integrating the joint density over y. As a function of y, the conditional density is proportional to the joint density at that x; dividing by fX(x) only rescales it so the area under the curve is 1 (Davis Ross). To see where the variable X comes from, read what a random variable is.
Worked example with real numbers
Discrete: a joint count table
The counts below are illustrative numbers invented for this guide, not data from a real study. 100 participants are classed by training format (X) and by a competency rating (Y) scored Low = 1, Mid = 2, High = 3.
| Training (X) | Low | Mid | High | Row total |
|---|---|---|---|---|
| None | 12 | 18 | 10 | 40 |
| Online | 9 | 21 | 20 | 50 |
| In-person | 5 | 3 | 2 | 10 |
| Column total | 26 | 42 | 32 | 100 |
Step 1, marginal of X. Sum each row: pX(None) = 40/100 = 0.40, pX(Online) = 0.50, pX(In-person) = 0.10.
Step 2, divide. Each cell over 100 is a joint probability (Online, High = 20/100 = 0.20). For X = Online: pY|X(High | Online) = 0.20 / 0.50 = 0.40, pY|X(Mid | Online) = 0.21 / 0.50 = 0.42, pY|X(Low | Online) = 0.09 / 0.50 = 0.18. These add to 0.18 + 0.42 + 0.40 = 1.00, as a distribution must.
| Conditional distribution of Y given X | P(Low) | P(Mid) | P(High) | Sum | E[Y | X] |
|---|---|---|---|---|---|
| None (n = 40) | 0.30 | 0.45 | 0.25 | 1.00 | 1.95 |
| Online (n = 50) | 0.18 | 0.42 | 0.40 | 1.00 | 2.22 |
| In-person (n = 10) | 0.50 | 0.30 | 0.20 | 1.00 | 1.70 |
Step 3, conditional mean. E[Y | X = Online] = 1(0.18) + 2(0.42) + 3(0.40) = 0.18 + 0.84 + 1.20 = 2.22. The same sum gives 1.95 for None (0.30 + 0.90 + 0.75) and 1.70 for In-person (0.50 + 0.60 + 0.60).
The reverse is a different distribution. P(High | Online) = 20/50 = 0.40, but P(Online | High) = 20/32 = 0.625. The first divides by the Online row total, the second by the High column total.
Continuous: slicing a joint density
This example follows Davis Ross, example 23.2. Spin two independent Uniform(1, 4) spinners; let X be the sum and Y the larger value. The joint density is fX,Y(x, y) = 2/9 on the region 1 < y < 4, y + 1 < x < 2y (the same region as x/2 < y < x − 1). Integrating over x gives the marginal fY(y) = (2/9)(2y − (y + 1)) = (2/9)(y − 1).
Condition on Y = 3. Then x runs from 4 to 6, fY(3) = (2/9)(2) = 4/9, and fX|Y(x | 3) = (2/9) / (4/9) = 1/2 for 4 < x < 6. That is a Uniform(4, 6) density, and its area is 2 × 1/2 = 1. So P(X > 5.5 | Y = 3) = (6 − 5.5) × 1/2 = 1/4.
Interpretation
Read each conditional distribution as an ordinary distribution for the subgroup defined by the condition. Compare rows (or slices) to see whether knowing X changes what you expect for Y. If every conditional distribution of Y is identical, X tells you nothing about Y; in the example above the three rows differ, so training format and rating are associated in these numbers.
| What you see | How to read it |
|---|---|
| Conditional probabilities sum to 1 across y | The slice was divided by the marginal correctly. |
| Rows differ from one another | The distribution of Y depends on X; the example’s rows range from E[Y | X] = 1.70 to 2.22. |
| A conditional density value above 1 | Allowed for a density; only areas are probabilities. |
| P(Y | X) differs from P(X | Y) | They answer different questions and use different denominators. |
Assumptions and common mistakes
- Conditioning on something impossible. The formulas need pX(x) > 0 (or fX(x) > 0). A row with no observations has no conditional distribution.
- Forgetting to divide. A raw slice of the joint table is not a distribution; in the example the Online row of joint probabilities sums to 0.50, not 1.
- Swapping the direction. P(High | Online) = 0.40 is not P(Online | High) = 0.625.
- Reading a density as a probability. fY|X(y | x) = 1/2 above is a height, not a 50% chance.
How to report it (APA example)
Name the conditioning variable, give the group sizes, and report the conditional proportions or the conditional mean. For the table above:
Among participants who completed online training (n = 50), 40% received a High rating, compared with 25% of participants with no training (n = 40) and 20% of those with in-person training (n = 10). The conditional mean rating was 2.22 for online training, 1.95 for no training and 1.70 for in-person training.
State the denominator for each percentage, and say which variable is the condition. Reporting a conditional proportion without its group size hides how much data it rests on. For related reporting of a hypothesis-test result, see what a p value is.
How to calculate it in Excel, R or Python
Excel
Put the counts in B2:D4 with scores 1, 2, 3 for the columns. In F2 enter =B2/SUM($B2:$D2) and fill across to H4 (adjust the column references as you fill) to get each conditional probability. For the conditional mean in I2: =SUMPRODUCT(B2:D2,{1,2,3})/SUM(B2:D2), filled down.
R
m <- matrix(c(12,18,10, 9,21,20, 5,3,2), nrow = 3, byrow = TRUE,
dimnames = list(c("None","Online","In-person"), c("Low","Mid","High")))
cond <- prop.table(m, 1) # divide each row by its row total
rowSums(cond) # 1 1 1
cond %*% 1:3 # conditional means: 1.95, 2.22, 1.70Python
import pandas as pd
df = pd.DataFrame([[12,18,10],[9,21,20],[5,3,2]],
index=["None","Online","In-person"], columns=["Low","Mid","High"])
cond = df.div(df.sum(axis=1), axis=0) # P(Y | X): rows sum to 1
means = (cond * [1, 2, 3]).sum(axis=1) # E[Y | X]
print(cond.round(2)); print(means.round(2))For the other direction, P(X | Y), divide by column totals instead: df.div(df.sum(axis=0), axis=1).
FAQ
What is a conditional probability distribution?
It is the distribution of one variable among the cases where another variable has a given value, equal to the joint distribution divided by the marginal of the condition.
How to calculate a probability value from it?
Add the conditional probabilities of the outcomes you want, or integrate the conditional density over the range you want. In the continuous example, P(X > 5.5 | Y = 3) = 1/4.
Is “probability value” the same as a p-value?
No. A p-value is a probability computed under a null hypothesis; see the p-value dictionary entry. This guide is about the distribution of one variable given another.
What is probability of default?
It is a finance term for the probability that a borrower fails to repay. Like any probability it can be conditioned on other information, but it is a single event probability rather than the full distribution described here.
Sources
- Davis Ross, K. STAT 350 handouts, “Conditional distributions” (handout 23), https://bookdown.org/kevin_davisross/stat350-handouts/conditional-pmf-and-pdf.html. Date checked: 29 October 2026.
- StatLect, “Conditional probability distributions”, https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions. Date checked: 29 October 2026.
