Probability of 3 Events

For Independent Events

Enter the probabilities of three independent events (A, B, and C) to calculate various resulting probabilities.

Examples

See how to use the calculator with these real-world scenarios.

Three Coin Flips

coin_flips

Calculate the probability of getting heads on three consecutive, independent coin flips.

P(A): 0.5

P(B): 0.5

P(C): 0.5

Email Campaign Success

email_campaign

An email campaign's success depends on three factors: deliverability (95%), open rate (25%), and click-through rate (10%). What's the probability of a single email resulting in a click?

P(A): 0.95

P(B): 0.25

P(C): 0.10

Manufacturing Defects

manufacturing_defects

Three independent machines produce a part. Machine A has a 2% defect rate, B has a 5% rate, and C has a 1% rate. What is the probability that a randomly selected part from the line is not defective (assuming one part from each machine)?

P(A): 0.98

P(B): 0.95

P(C): 0.99

Sports Parlay Bet

sports_betting

A bettor places a parlay on three games. The implied probabilities of winning each bet are 60%, 70%, and 50%. What's the probability of winning the entire parlay?

P(A): 0.60

P(B): 0.70

P(C): 0.50

Other Titles
Understanding the Probability of 3 Events
A Comprehensive Guide to Calculating Probabilities for Multiple Independent Events

What is the Probability of Multiple Events?

  • Core Concepts of Probability
  • Independent vs. Dependent Events
  • The Multiplication Rule for Independent Events
In probability theory, we often want to understand the likelihood of multiple events occurring. The calculation method depends critically on whether the events are independent or dependent. Independent events are those where the outcome of one does not affect the outcome of the others. A classic example is flipping a coin multiple times. This calculator focuses exclusively on these types of independent events.
The Foundation: Independent Events
For three independent events A, B, and C, the probability that all three occur is found by simply multiplying their individual probabilities. This is known as the multiplication rule.
P(A and B and C) = P(A) × P(B) × P(C)

Example

  • If the probability of rain is 0.3 (P(A)), the probability of a traffic jam is 0.5 (P(B)), and the probability of your favorite song playing is 0.1 (P(C)), the chance of all three happening (assuming they are independent) is 0.3 * 0.5 * 0.1 = 0.015 or 1.5%.

Step-by-Step Guide to Using the Calculator

  • Inputting Your Probabilities
  • Interpreting the 'AND' and 'OR' Results
  • Understanding 'Exactly One/Two' and 'None' Outcomes
1. Enter the Probability of Event A, P(A)
In the first input field, enter the probability of the first event (A) occurring. This must be a decimal value between 0 (the event is impossible) and 1 (the event is certain).
2. Enter P(B) and P(C)
Repeat the process for the second and third events, P(B) and P(C), in their respective fields.
3. Click 'Calculate'
The calculator will instantly provide five key results based on your inputs.

Mathematical Derivations and Formulas

  • Formula for P(A and B and C)
  • Formula for P(A or B or C)
  • Formulas for Complex Scenarios
The results are calculated using fundamental probability rules. Let P(A), P(B), and P(C) be the probabilities of the three events. Let P(A'), P(B'), and P(C') be the probabilities of them not occurring (e.g., P(A') = 1 - P(A)).
Probability of All Events Occurring:
P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
Probability of None of the Events Occurring:
P(A' ∩ B' ∩ C') = (1 - P(A)) × (1 - P(B)) × (1 - P(C))
Probability of At Least One Event Occurring (A or B or C):
P(A ∪ B ∪ C) = 1 - P(A' ∩ B' ∩ C')
Probability of Exactly One Event Occurring:
P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C)
Probability of Exactly Two Events Occurring:
P(A ∩ B ∩ C') + P(A ∩ B' ∩ C) + P(A' ∩ B ∩ C)

Real-World Applications of 3-Event Probability

  • Risk Assessment in Finance and Insurance
  • Quality Control in Manufacturing
  • System Reliability and Engineering
Engineering and System Redundancy
Engineers use these calculations to determine the reliability of a system. If a critical component has two backup systems, what is the probability that all three fail? You would input the failure probability of each (P(A), P(B), P(C)) to find P(A and B and C), which represents a total system failure.
Medical Diagnosis
Imagine a patient is tested for three independent risk factors for a disease. A doctor can calculate the probability of the patient having all three risk factors, or at least one, to better assess their overall health risk.

Common Misconceptions

  • Confusing Independent and Dependent Events
  • The Gambler's Fallacy
  • Adding Probabilities Incorrectly
The 'AND' vs. 'OR' Confusion
A common mistake is adding probabilities when multiplication is needed. The probability of 'A and B' happening is always lower than the individual probabilities (since it's a more specific outcome), so we multiply. The probability of 'A or B' happening is higher (a less specific outcome), which involves addition (and subtraction of the overlap).
Ignoring the Independence Assumption
This calculator's formulas are only valid if the events are independent. If event A makes event B more or less likely, the events are dependent, and more complex formulas involving conditional probability are required. For example, the event 'it is cloudy' and 'it will rain' are dependent; the formulas here would not apply.