Calculator
Formula used
P(A ∩ B) = P(A) × P(B | A)
P(B | A) = P(A ∩ B) / P(A)
For count-based problems, P(A) = favorable outcomes for A / total outcomes.
For changing sample spaces, P(B | A) = favorable outcomes for B after A / remaining outcomes after A.
When one result changes the next result, the events are dependent.
How to use this calculator
Choose a calculation mode first.
Use counts mode for cards, marbles, balls, and selections without replacement.
Enter the total outcomes and the favorable outcomes for the first event.
Then enter the favorable outcomes and remaining outcomes for the second event after the first event happens.
Use direct mode when you already know P(A) and P(B | A).
Set your preferred decimal precision, then press the calculate button.
The result box appears above the form and shows formulas, decimals, percentages, and export options.
Example data table
| Scenario | P(A) | P(B | A) | P(A ∩ B) |
|---|---|---|---|
| Pick a red marble, then a blue marble, without replacement | 4/10 | 3/9 | 12/90 = 2/15 |
| Draw an ace, then a king, without replacement | 4/52 | 4/51 | 16/2652 |
| Select a math book, then a science book, without replacement | 6/20 | 5/19 | 30/380 = 3/38 |
Understanding dependent event probability
What dependent events mean
Dependent events do not stand alone. The first result changes the second result. That change affects the sample space. It also changes the probability of the next event. A common example is drawing cards without replacement. After one card leaves the deck, both the total cards and the category counts can change.
Why conditional probability matters
Conditional probability is the center of dependent event questions. It asks for the chance of one event after another event already happened. In notation, that is P(B | A). The vertical bar means “given.” Once event A occurs, the second event must be measured from the updated sample space. This is why dependent probability problems are different from independent ones.
How the joint probability is found
The probability of both events happening together is called joint probability. For dependent events, use the rule P(A ∩ B) = P(A) × P(B | A). The first factor measures the chance of the opening event. The second factor measures the chance of the follow-up event under the new condition. This method works well for urn models, card draws, seating tasks, and many classroom examples.
Why this calculator helps
This calculator supports two strong methods. You can enter raw counts for changing sample spaces. You can also enter direct probability values. The result section shows decimals, percentages, and step-by-step logic. That makes checking homework easier. It also builds a better understanding of event sequences, conditional thinking, and probability notation. Use it to compare scenarios, test class examples, and verify manual solutions with confidence.
Frequently asked questions
1. What are dependent events?
Dependent events are events where the first outcome changes the probability of the next outcome. The sample space shifts after the first event happens.
2. What does P(B | A) mean?
P(B | A) means the probability of event B given that event A has already occurred. It is a conditional probability.
3. Why does drawing without replacement create dependence?
Without replacement, one item is removed after the first draw. That changes the total number of items and often changes the favorable count too.
4. When should I use counts mode?
Use counts mode when you know totals and favorable outcomes. It is ideal for marbles, cards, tickets, books, and other selection problems.
5. When should I use direct probability mode?
Use direct mode when your problem already gives P(A) and P(B | A). Then the calculator multiplies them to find the joint probability.
6. Does order matter in dependent events?
Yes. Order often matters because the second event is measured after the first event. Reversing the order can produce a different answer.
7. What is joint probability?
Joint probability is the chance that both events happen together. For dependent events, multiply the first event probability by the conditional probability.
8. Can this calculator show percentages and decimals?
Yes. The result table shows both decimal values and percentage values. It also includes step-by-step working for easier checking.