Calculator Form
Example Data Table
| Scenario | P(A) | P(B|A) | P(A and B) | Interpretation |
|---|---|---|---|---|
| Draw a red marble, then a blue marble, without replacement | 5/8 = 0.6250 | 3/7 = 0.4286 | 15/56 = 0.2679 | The second event depends on the first draw. |
| Select a passing student, then one who completed homework, from an updated group | 0.70 | 0.80 | 0.56 | The joint probability is 56%. |
| Inspect a machine cycle, then check a follow-up quality pass after that cycle | 0.92 | 0.95 | 0.8740 | The follow-up result uses a conditional chance. |
Formula Used
Two dependent events:
P(A and B) = P(A) × P(B|A)
This formula multiplies the probability of the first event by the conditional probability of the second event after the first event occurs.
Three dependent events:
P(A and B and C) = P(A) × P(B|A) × P(C|A and B)
Each added event uses the updated condition created by the earlier events.
Complement:
P(not joint event) = 1 − P(joint event)
Expected count:
Expected count = Sample size × Joint probability
How to Use This Calculator
- Enter labels for your events if you want custom names.
- Choose decimal mode or percentage mode.
- Enter the probability of the first event, P(A).
- Enter the conditional probability of the second event, P(B|A).
- Add P(C|A and B) if you want a three-event chain.
- Enter a sample size to estimate expected occurrences.
- Select the number of decimal places you want.
- Press Calculate to show results above the form.
- Use the CSV button for spreadsheet data.
- Use the PDF button to save the result block.
Dependent Events in Probability
Dependent events happen when one outcome changes another outcome. This idea is common in maths, statistics, quality control, and real decision work. A second draw without replacement is a classic example. The first draw changes what remains. That change affects the next probability. This calculator helps you measure that linked effect quickly. It works for two events or a three-event chain. You can also switch between decimals and percentages. That makes it useful for school tasks, reports, and fast probability checks.
Why Conditional Probability Matters
Conditional probability tells you how likely an event is after another event has already happened. That is why dependent events cannot use a simple independent rule. You need the updated chance, not the original one. In this calculator, you enter P(A) and then P(B|A). The tool multiplies them to produce the joint probability. If you add a third event, it continues the chain correctly. This structure matches real sequences where each step depends on previous results, such as sampling, testing, screening, and staged approvals.
How to Read the Output
The result section shows the decimal form and percentage form of each key value. It also shows the complement. That is the chance the full event chain does not happen. The one-in-N estimate gives a simple frequency view. The expected count uses your sample size. This is useful when you want to estimate how many joint outcomes may appear in repeated trials. These extra outputs make the calculator more practical than a basic formula display. They help you explain results with clear context.
Where This Calculator Helps
You can use a dependent events calculator for urn problems, card draws, quality checks, medical screening paths, and classroom probability lessons. It is also helpful for process planning. Many real systems follow linked steps. Each step changes the next one. By using conditional inputs, this calculator keeps the maths accurate. The example table, formula guide, and FAQ section also support learning. That makes the page suitable for both quick calculation and concept review. It combines clarity, structure, and flexible probability analysis in one place.
FAQs
1. What are dependent events?
Dependent events are events where one outcome changes the probability of the next outcome. Drawing without replacement is the most common example.
2. How is this different from independent events?
Independent events do not affect each other. Dependent events do. This calculator uses conditional probability, which is required when earlier outcomes change later chances.
3. What does P(B|A) mean?
P(B|A) means the probability of event B happening after event A has already happened. It is a conditional probability.
4. Can I enter percentages instead of decimals?
Yes. Choose percentage mode and enter values like 40 or 75. The calculator converts them internally before computing the joint probability.
5. What is the complement result?
The complement is the chance that the full joint event does not occur. It is found by subtracting the joint probability from 1.
6. Why is sample size included?
Sample size helps estimate expected occurrences. If the joint probability is 0.20 and the sample size is 100, the expected count is 20.
7. Can I calculate three dependent events?
Yes. Enter the optional value for P(C|A and B). The calculator will extend the probability chain and show the three-event result.
8. When should I use this calculator?
Use it for maths homework, probability lessons, quality scenarios, sequential testing, sampling problems, or any case where one event changes the next event.