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Calculate the probability of single events, compound events (AND/OR), and conditional probability. Get instant results with step-by-step explanations, odds representation, and fraction conversion.
P(A) = favorable outcomes / total outcomesThe probability of a single event equals favorable outcomes divided by total possible outcomes.
This probability calculator is for educational purposes. Results are based on theoretical probability calculations.
For real-world applications, consider factors like experimental probability, sample size, and statistical significance.
Probability is a measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It can also be expressed as a percentage (0% to 100%) or as odds. Probability is fundamental to statistics, data science, gambling, insurance, weather forecasting, and countless other fields. Understanding probability helps us make informed decisions under uncertainty.
The basic probability formula is: P(A) = Number of favorable outcomes / Total number of possible outcomes For example, the probability of rolling a 6 on a fair die is 1/6 because there is 1 favorable outcome (rolling a 6) out of 6 possible outcomes (1, 2, 3, 4, 5, or 6). Probabilities always sum to 1 when considering all possible outcomes. The complement of an event (the probability it does NOT happen) equals 1 minus the probability it does happen.
There are several types of probability: 1. Simple (Single Event) Probability: The likelihood of one event occurring, like flipping heads on a coin. 2. Compound Probability (AND): The probability that two or more events all occur together. For independent events: P(A and B) = P(A) x P(B). 3. Compound Probability (OR): The probability that at least one of several events occurs. P(A or B) = P(A) + P(B) - P(A and B). 4. Conditional Probability: The probability of an event given that another event has occurred. P(A|B) = P(A and B) / P(B).
Key probability formulas: Single Event: P(A) = favorable outcomes / total outcomes Complement: P(not A) = 1 - P(A) AND (Independent): P(A and B) = P(A) x P(B) AND (Dependent): P(A and B) = P(A) x P(B|A) OR (General): P(A or B) = P(A) + P(B) - P(A and B) OR (Mutually Exclusive): P(A or B) = P(A) + P(B) Conditional: P(A|B) = P(A and B) / P(B)
Example 1: What is the probability of drawing a heart from a standard deck? Answer: 13 hearts / 52 cards = 1/4 = 0.25 = 25% Example 2: What is the probability of flipping two heads in a row? Answer: P(H and H) = 0.5 x 0.5 = 0.25 = 25% Example 3: What is the probability of rolling a 1 or a 6 on a die? Answer: P(1 or 6) = 1/6 + 1/6 = 2/6 = 1/3 = 33.33%
1. Always identify whether events are independent or dependent before calculating. 2. Remember that probabilities must be between 0 and 1 (or 0% and 100%). 3. For 'AND' problems, multiply probabilities. For 'OR' problems, add them (but subtract the overlap if events aren't mutually exclusive). 4. Use tree diagrams or tables to visualize complex probability problems. 5. Check your answer: the sum of all possible outcomes should equal 1.
Probability measures the chance of an event occurring (favorable outcomes / total outcomes). Odds compare favorable outcomes to unfavorable outcomes. For example, if the probability is 1/4, the odds are 1:3 (1 favorable to 3 unfavorable).
Independent events don't affect each other - like flipping a coin twice. The first flip doesn't change the probability of the second. Dependent events do affect each other - like drawing cards without replacement. After drawing one card, the probabilities for the next draw change.
Use the addition rule: P(A or B) = P(A) + P(B) - P(A and B). If events are mutually exclusive (can't both happen), then P(A and B) = 0, so P(A or B) = P(A) + P(B).
Conditional probability, written as P(A|B), is the probability of event A occurring given that event B has already occurred. It's calculated as P(A|B) = P(A and B) / P(B).
Because an event either happens or it doesn't - there are no other possibilities. If P(rain) = 0.3, then P(no rain) = 0.7. Together, 0.3 + 0.7 = 1, which accounts for 100% of possible outcomes.
To convert probability to odds: If P(A) = p, then odds for A are p:(1-p). For example, if P(win) = 0.25, the odds are 0.25:0.75, which simplifies to 1:3.
The multiplication rule calculates P(A and B). For independent events: P(A and B) = P(A) x P(B). For dependent events: P(A and B) = P(A) x P(B|A).
No. Probability is always between 0 and 1 (inclusive). A probability of 0 means impossible, 1 means certain. If your calculation gives a result greater than 1, check your inputs and formula.