Two jars with colored beads and a small transfer spoon. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
The second draw sees a changed pool
A bag contains five red and five blue tokens. Draw two tokens without replacement. The probability of two reds is (5/10)(4/9) = 2/9, about 22.2222%. Multiplying 1/2 by 1/2 would describe a different experiment in which the first token is returned before the second draw.
Label the conditional factor
Let A mean first draw red and B mean second draw red. The general multiplication rule is P(A and B) = P(A)P(B | A). Here the second factor is 4/9 because the first red draw leaves four reds among nine tokens. The unconditional probability of red on the second draw is still one-half by symmetry, but that is not the conditional probability needed for the joint event.
Check both possible orders
For red then blue, the product is (5/10)(5/9) = 5/18. Blue then red has the same probability. If the event is one red and one blue in either order, add the two disjoint ordered events to get 5/9. Do not add them when the event specifies red first: that would expand the event without saying so.
Match the input mode to the experiment
Use the Joint Probability Calculator in marginal-and-conditional mode with 50% and approximately 44.4444444% for the two-red example. Its independent mode is appropriate only when the experiment justifies independence. If observations already form a table, the joint-count guide explains the denominator. The existing Conditional Probability Calculator instead starts from compatible marginals and a joint probability to reverse the conditional direction.
Check the model behind your result
Use the Joint Probability Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26