Probability Calculator
Calculate probability for single and combined events. Enter favorable and total outcomes.
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Understanding Probability
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). The probability calculator handles single events, complements, and combined events for independent outcomes.
Basic Probability Formula
P(A) = (number of favorable outcomes) / (total number of equally likely outcomes). Rolling a 3 on a die: P = 1/6 ≈ 0.167. Drawing a heart from a deck: P = 13/52 = 0.25. The result can be expressed as a fraction, decimal, or percentage.
Complement Rule
The complement of event A is "A does not happen": P(A') = 1 - P(A). If there's a 30% chance of rain, there's a 70% chance it won't rain. The complement rule is fundamental to probability and helps when calculating "at least one" scenarios.
Independent Events: P(A and B)
For two independent events (the outcome of one doesn't affect the other): P(A and B) = P(A) × P(B). Flipping heads and rolling a 6: P = 0.5 × (1/6) ≈ 0.083. Independence is key — if events are dependent, a conditional probability formula is needed instead.
Union: P(A or B)
P(A or B) = P(A) + P(B) - P(A and B). The subtraction prevents double-counting events where both occur. For mutually exclusive events (can't both happen), P(A and B) = 0, so P(A or B) = P(A) + P(B).
Expressing Probability
Probability can be expressed as a fraction (1/4), decimal (0.25), percentage (25%), or odds (3 to 1 against). Odds express the ratio of unfavorable to favorable outcomes: if P = 0.25, odds against = 3:1, meaning for every 3 failures you'd expect 1 success on average.
Worked Example: Card Drawing
Draw one card from a 52-card deck. Event A: drawing a heart (13 hearts / 52 total = 0.25 or 25%). Event B: drawing a face card (J, Q, K — 12 face cards / 52 = 0.231 or 23.1%). P(heart AND face card) = 3/52 ≈ 0.0577 (three face cards are hearts). P(heart OR face card) = 0.25 + 0.231 − 0.0577 = 0.423 or 42.3%. The subtraction removes the double-counted heart face cards.
| Rule | Formula | Example (die: P(even)=0.5, P(>4)=1/3) |
|---|---|---|
| Single event | P(A) = favourable/total | P(even) = 3/6 = 0.5 |
| Complement | 1 − P(A) | P(not even) = 0.5 |
| AND (independent) | P(A) × P(B) | P(even AND >4) = 0.5 × 0.333 = 0.167 |
| OR (general) | P(A) + P(B) − P(A and B) | P(even OR >4) = 0.5+0.333−0.167 = 0.667 |
Frequently Asked Questions
Probability is a number between 0 and 1 measuring how likely an event is. P(A) = favorable outcomes ÷ total outcomes. 0 = impossible, 1 = certain, 0.5 = equally likely either way.
P(A') = 1 - P(A). If P(rolling a 6) = 1/6, then P(not rolling a 6) = 5/6 ≈ 0.833.
Multiply: P(A and B) = P(A) × P(B). Rolling a 6 twice: (1/6) × (1/6) = 1/36 ≈ 0.028 (about 2.8%).
P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events (can't both occur), just P(A) + P(B).
Probability: favorable / total. Odds in favor: favorable / unfavorable. If P = 0.25, odds in favor = 1:3, odds against = 3:1.
Events A and B are independent if the occurrence of A doesn't change the probability of B. Coin flips are independent; drawing cards without replacement are not (dependent).
Mutually exclusive events cannot both occur at the same time (rolling a 1 and rolling a 6 on one die). Independent events can both occur — one doesn't affect the other's probability (coin flip and die roll). Mutually exclusive events are actually dependent: if A occurs, P(B) drops to zero.
Use the complement: P(at least one) = 1 − P(none). For 5 coin flips, P(at least one head) = 1 − P(all tails) = 1 − (0.5)⁵ = 1 − 0.03125 = 0.96875. This is much easier than adding up the probabilities of exactly 1, 2, 3, 4, and 5 heads.