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To calculate conditional probability, divide the probability that both events happen by the probability of the event you are conditioning on: P(A|B) = P(A and B) / P(B). The result is the chance of A once you know B has occurred. This guide works one two-way table through every step, including the swapped-conditional mistake, an independence check, and Excel, R and Python code.
What is conditional probability?
Conditional probability is the probability that an event A occurs given that another event B has already occurred. It is written P(A|B) and read “the probability of A given B” (OpenStax Introductory Statistics 2e, section 3.1). Knowing B happened shrinks the sample space to the outcomes in B, and A is then measured only inside that smaller space. For a die roll, the sample space {1, 2, 3, 4, 5, 6} reduces to {1, 3, 5} when you are told the roll is odd (OpenStax, section 3.2).
Conditional probability formula: P(A|B) = P(A and B) / P(B)
The formula is the multiplication rule rearranged. OpenStax states the multiplication rule as P(A and B) = P(B)P(A|B); dividing both sides by P(B) gives the conditional form (section 3.3).
- P(A|B): probability of A given B. This is the number you want.
- P(A and B): probability that A and B both occur (the joint probability).
- P(B): probability of the conditioning event. It must be greater than 0, because you cannot condition on an event that never happens.
The same idea works with counts. In a table, P(A|B) is the count in the cell where A and B overlap divided by the total count for B. The joint probability is covered separately in the joint probability formula guide.
How to calculate conditional probability step by step
- Define the event you are conditioning on (B) and the event you want (A). The word “given”, “if” or “among” usually marks B.
- Find P(B), or the total count for B.
- Find P(A and B), or the count where both hold.
- Divide the joint value by the value for B.
- Check that the answer lies between 0 and 1 and that you divided by the right total.
Worked example from a two-way table
The table below is the 100-person handedness example used in OpenStax section 3.1.
| Right-handed | Left-handed | Total | |
|---|---|---|---|
| Male | 43 | 9 | 52 |
| Female | 44 | 4 | 48 |
| Total | 87 | 13 | 100 |
Probability of left-handed given female
Let L = left-handed and F = female. The condition is F, so the sample space becomes the 48 females.
- P(L and F) = 4/100 = 0.04
- P(F) = 48/100 = 0.48
- P(L|F) = 0.04 / 0.48 = 0.0833
Counting directly gives the same answer: 4 left-handed females out of 48 females is 4/48 = 0.0833, which matches OpenStax.
Why the order matters: P(F|L) is not P(L|F)
Reverse the condition and the denominator changes. Given left-handed, the sample space is the 13 left-handed people: P(F|L) = 4/13 = 0.3077. Both answers share the numerator 4 and differ only in the denominator (48 versus 13). Confusing the two is the most common error in conditional probability.
Reversing the condition with Bayes’ rule
If you know P(L|F), P(F) and P(L), you can reverse the condition: P(F|L) = P(L|F) × P(F) / P(L) = 0.0833 × 0.48 / 0.13 = 0.04 / 0.13 = 0.3077, the same value as the direct count. For the Bayesian framing, see interpreting Bayes factors and Bayesian vs frequentist statistics.
Interpreting the result
| Comparison | What it tells you |
|---|---|
| P(A|B) = P(A) | Knowing B does not change the probability of A, so A and B are independent (OpenStax, section 3.2). |
| P(A|B) > P(A) | B makes A more likely in this data. |
| P(A|B) < P(A) | B makes A less likely in this data. |
In the table, P(L) = 13/100 = 0.13 while P(L|F) = 0.0833, so being female lowers the proportion of left-handers in this sample, and L and F are not independent here.
How to check independence with conditional probability
OpenStax gives three equivalent conditions for independence: P(A|B) = P(A), P(B|A) = P(B), or P(A and B) = P(A)P(B); you need to show only one (section 3.2). Its Example 3.9 has P(G) = 0.6, P(H) = 0.5 and P(G and H) = 0.3. Then P(G|H) = 0.3 / 0.5 = 0.6 = P(G), so G and H are independent, and the product check agrees: 0.6 × 0.5 = 0.3. OpenStax also advises that if you do not know whether events are independent, you should assume they are dependent until you can show otherwise.
Assumptions and common mistakes
- P(B) must be positive. The formula divides by P(B).
- Swapped conditions. P(A|B) and P(B|A) usually differ (0.0833 versus 0.3077 above).
- Wrong denominator. Divide by the total for the condition, not by the grand total. Using 4/100 instead of 4/48 gives the joint probability, not the conditional one.
- Assuming independence. Multiplying P(A) × P(B) for a joint probability is valid only when independence has been shown.
- Treating a sample proportion as a population probability. The 0.0833 describes these 100 people.
How to report conditional probability
Name both events, give the counts and the proportion, and state the base used. Example wording for a results section:
Among the 48 female participants, 4 were left-handed, a conditional probability of P(L|F) = .08, compared with an overall left-handed proportion of .13 (13 of 100).
Give the counts so readers can recompute the figure, and say which group is the denominator.
How to calculate conditional probability in Excel, R or Python
Excel
Put the counts in B2:C3 (rows Male and Female, columns Right and Left). Then P(L|F):
=C3/SUM(B3:C3)This returns 0.083333.
R
tab <- matrix(c(43, 44, 9, 4), nrow = 2,
dimnames = list(sex = c("M", "F"), hand = c("R", "L")))
prop.table(tab, 1)["F", "L"] # row proportions: P(L|F) = 0.0833prop.table(tab, 1) divides each cell by its row total, which is conditioning on the row variable. Use prop.table(tab, 2) to condition on columns and get P(F|L) = 0.3077.
Python (pandas)
import pandas as pd
df = pd.DataFrame({"R": [43, 44], "L": [9, 4]}, index=["M", "F"])
p_l_given_f = df.loc["F", "L"] / df.loc["F"].sum() # 0.0833FAQ
How do you calculate conditional probability?
Divide the probability that both events occur by the probability of the given event: P(A|B) = P(A and B) / P(B). With a count table, divide the cell count by the total for the given event, as in 4/48 = 0.0833 above.
What is the difference between conditional and joint probability?
A joint probability, P(A and B), is measured against everyone (4/100 = 0.04). A conditional probability is measured only within B (4/48 = 0.0833).
How do you know if two events are independent?
If P(A|B) = P(A), knowing B tells you nothing about A, and the events are independent.
Sources
- Illowsky, B., & Dean, S. Introductory Statistics 2e, OpenStax. 3.1 Terminology; 3.2 Independent and Mutually Exclusive Events; 3.3 Two Basic Rules of Probability. Date checked: 2026-10-29.
