Compute basic probability, conditional probability, combinatorics, odds, birthday paradox & expected value
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Probability is the mathematical language of uncertainty, governing everything from gambling odds to scientific predictions to everyday decision-making. This probability calculator brings together six essential probability tools in one place, eliminating the need to switch between multiple calculators or work through tedious manual computations.
Whether you’re a student studying statistics, a teacher preparing classroom examples, a gambler analyzing odds, or a professional working with risk assessment, this tool adapts to your needs. Choose from six specialized modes: basic probability (favorable over total outcomes), conditional probability using Bayes’ theorem, combinatorics for counting arrangements and selections, odds conversion between American, decimal, and fractional formats, the surprising birthday paradox, and expected value calculations for decision analysis.
How to Use the Probability Calculator
Select your calculation mode by clicking one of the mode tabs: Basic, Conditional, Combinatorics, Odds, Birthday, or Expected Value.
Enter your input values in the fields displayed for your chosen mode, each tab shows only the inputs needed for that calculation type.
Watch results update instantly as you type, or click the “Calculate” button where provided for certain modes.
Try any quick example button to see how the tool works with common scenarios like coin flips, dice rolls, or card draws.
Click the “Copy” button to copy your current results to your clipboard with a single tap.
Use the “Share” button to share your calculation via your device’s sharing options.
Reset everything with the “Reset All” button to return all modes to their default values.
Formula:
The simplest probability calculation divide the number of favorable outcomes by the total number of possible outcomes. Results show the probability as a decimal, a “1 in X” chance, and a percentage.
Formula:
This calculates the probability of event A occurring given that event B has occurred. Enter P(A), P(B), and P(B|A) to find P(A|B). The “1 in X” result shows the odds implied by the calculated probability.
Combinations (order doesn’t matter):
Permutations (order matters):
nPr = n! / (n-r)!
Combinations count selections where order doesn’t matter — choosing a committee from a group. Permutations count arrangements where order matters — assigning positions or ranking items.
Odds represent the relationship between probabilities of an event occurring versus not occurring. This tool converts between American odds (positive or negative numbers), decimal odds (quotients), and fractional odds (ratios), while also showing the implied probability.
Formula:
This calculates the probability that at least two people in a group share a birthday. The result is often surprisingly high — with just 23 people, there’s a 50% chance of a shared birthday.
Formula:
Expected value represents the long-term average outcome of a random variable. Multiply each outcome value by its probability, sum the results, and you get the expected value. The tool also calculates variance and standard deviation.
Suppose you want to calculate the probability of drawing a red card from a standard deck of 52 playing cards.
Step 1: Identify your values: Favorable outcomes = 26 (13 hearts + 13 diamonds), Total outcomes = 52 cards.
Step 2: Apply the basic probability formula: P(red) = 26 / 52
Step 3: Simplify: 26/52 = 0.5
Step 4: Interpret the result: You have a 0.5 probability (50%) of drawing a red card. This means in 1 in 2 draws, you’d expect to get a red card.
What this means: If you repeatedly draw one card from a shuffled deck and return it, about half of your draws would be red cards. This forms the foundation for more complex probability calculations like conditional probabilities when multiple cards are drawn without replacement.
Basic probability calculates the likelihood of a single event based on favorable outcomes over total outcomes. Conditional probability calculates the likelihood of one event given that another event has already occurred, essential for situations where events influence each other.
Use combinations when the order of selection doesn’t matter, choosing team members, selecting lottery numbers, or picking a subset of items. Use permutations when order matters assigning positions, ranking items, or arranging objects in sequence.
Probability is the chance an event occurs, expressed as a fraction between 0 and 1. Odds represent the ratio of probability of occurrence to probability of non-occurrence. Odds can be expressed as American (moneyline), decimal (multipliers), or fractional (ratios).
The birthday paradox shows that in a group of just 23 people, there’s a 50% chance two share a birthday. The surprise comes from our intuition failing to account for the number of possible pairs (253 pairs from 23 people) compared to 365 days.
Expected value gives the long-term average outcome of a repeated random process. For a bet, it’s what you’d win on average per trial. For decision-making, it helps compare options where outcomes and probabilities vary.
Yes, this tool uses advanced arithmetic for combinatorics to handle large numbers precisely. Birthday calculations and expected value are handled with standard floating-point arithmetic suitable for typical use cases.
Bayes’ theorem requires three inputs: prior probability P(A), evidence probability P(B), and likelihood P(B|A). The result P(A|B) gives the updated probability after considering the evidence. This is fundamental for medical testing, risk assessment, and machine learning.
Odds are derived from probability: odds = P / (1-P). Higher probability means higher odds. Odds conversion helps understand betting markets, where odds are typically quoted in different formats across different platforms.
Bayes’ theorem is a specific application of conditional probability that relates P(A|B) to P(B|A). It’s used when you know the probability of evidence given a condition but need the probability of the condition given the evidence.
In theory, yes a probability of 0 means impossible, 1 means certain. In practice, many probabilities approach these limits but aren’t exact due to real-world complexity. The coin landing on its edge is a classic example of near-zero but not zero probability.
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