Probability Calculator

Calculate the probability of any event - single, combined, or as permutations and combinations.

🎲 Probability Calculator

📖 What is Probability?

Probability is a branch of mathematics that quantifies how likely an event is to occur. It is expressed as a number between 0 (impossible) and 1 (certain), or equivalently as a percentage between 0% and 100%. The probability of flipping a fair coin and getting heads is 0.5 or 50%.

Probability theory underpins many real-world disciplines: insurance companies use it to price risk, weather services use it to forecast rain, casinos use it to design games, medical researchers use it in clinical trials, and machine learning uses it in classification algorithms.

There are three key types of probability problems:

1. Single event probability - what is the chance of one specific outcome from a set of possible outcomes? 2. Multiple event probability - what is the chance of A AND B both happening, or A OR B happening? 3. Combinatorial probability - over how many ways can you choose r items from n total, and what is the probability of any specific selection?

Understanding the difference between "and" (intersection) and "or" (union) is crucial. "P(A and B)" requires both to happen simultaneously; "P(A or B)" requires at least one to happen.

📐 Formula

Single Event:
P(A) = Favourable Outcomes / Total Outcomes
P(not A) = 1 - P(A)
Multiple Events (independent):
P(A and B) = P(A) × P(B)
P(A or B) = P(A) + P(B) - P(A and B)
Combinations: nCr = n! / (r! × (n−r)!)
Permutations: nPr = n! / (n−r)!

📖 How to Use This Calculator

1
Single Event: Enter favourable outcomes and total outcomes → get probability as fraction, decimal, and percentage.
2
Multiple Events: Enter P(A) and P(B), select whether events are independent or mutually exclusive.
3
nCr/nPr: Enter n and r to count combinations and permutations.

💡 Example Calculations

Example 1 - Rolling a dice

1
Probability of rolling a 4 on a standard six-sided die:
2
Favourable = 1, Total = 6 → P = 1/6 = 0.1667 = 16.67%
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Example 2 - Two events

1
P(rain today) = 0.40, P(traffic jam) = 0.60. Both are independent events.
2
P(rain AND traffic) = 0.40 × 0.60 = 0.24 = 24%
3
P(rain OR traffic) = 0.40 + 0.60 - 0.24 = 0.76 = 76%
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Example 3 - Lottery

1
How many ways to choose 6 numbers from 45?
2
45C6 = 45! / (6! × 39!) = 8,145,060 combinations
3
Probability of winning = 1/8,145,060 = 0.0000123%
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Frequently Asked Questions

What is probability?+
Probability is a number between 0 and 1 that expresses how likely an event is to occur. A probability of 0 means the event is impossible; 1 means it is certain. A probability of 0.5 means the event has a 50% chance of occurring.
What is the difference between theoretical and experimental probability?+
Theoretical probability is calculated from known outcomes (e.g., a fair coin has a 50% chance of heads). Experimental probability is measured from actual trials (e.g., if you flip a coin 100 times and get 47 heads, the experimental probability is 47%). With enough trials, experimental approaches theoretical.
What does P(A and B) mean?+
P(A and B) is the probability that both event A and event B occur. For independent events, P(A and B) = P(A) × P(B). For example, rolling a 6 on a dice AND flipping heads on a coin = 1/6 × 1/2 = 1/12.
What does P(A or B) mean?+
P(A or B) is the probability that at least one of A or B occurs. For mutually exclusive events: P(A or B) = P(A) + P(B). For non-exclusive events: P(A or B) = P(A) + P(B) - P(A and B).
What is conditional probability?+
Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. Formula: P(A|B) = P(A and B) / P(B). For example, the probability of drawing a king given that the first card drawn was a face card.
What is the difference between independent and dependent events?+
Two events are independent if the outcome of one does not affect the probability of the other. Example: flipping a coin twice - getting heads on the first flip does not change the probability of heads on the second flip (still 50%). Events are dependent when the first outcome changes the probability of the second. Example: drawing cards without replacement - after drawing an ace from a 52-card deck, the probability of drawing another ace on the next draw changes from 4/52 to 3/51.
What is the complement rule in probability?+
The complement rule states that P(event happening) + P(event not happening) = 1. Therefore: P(not A) = 1 - P(A). This is useful when it is easier to calculate the probability of something NOT happening. Example: probability of rolling at least one six in 4 rolls of a die. Direct calculation is complex, but: P(no six in 4 rolls) = (5/6)^4 = 0.482. So P(at least one six) = 1 - 0.482 = 0.518.
How do you calculate the probability of multiple events all happening?+
For independent events, multiply the individual probabilities. P(A and B) = P(A) x P(B). Example: probability of rolling a 6 twice in a row = 1/6 x 1/6 = 1/36 = approximately 2.78%. For dependent events, use conditional probability: P(A and B) = P(A) x P(B given A). Example: probability of drawing 2 aces from a deck without replacement = 4/52 x 3/51 = 12/2652 = approximately 0.45%.
Tags: Probability Chance Likelihood Odds Coin Flip Dice Permutation Combination Events P(A and B) P(A or B)

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📌 Quick Tips

💡Probability always ranges from 0 (impossible) to 1 (certain). A probability of 0.5 means a 50% chance.
💡For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
💡The complement rule: P(not A) = 1 - P(A). It's often easier to calculate the probability that something doesn't happen.