Probability Calculator

Probability measures how likely an event is to occur. It is expressed as a number between $0$ and $1$ (or equivalently, $0\%$ to $100\%$).

$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Key Concepts

• Sample space $S$: the set of all possible outcomes
• Event $A$: a subset of the sample space
• Complement $A'$: the event that $A$ does NOT occur; $P(A') = 1 - P(A)$

Types of Probability

• Theoretical probability: Based on reasoning about equally likely outcomes (e.g., a fair coin has $P(\text{heads}) = \frac{1}{2}$)
• Experimental probability: Based on observed frequencies from experiments
• Subjective probability: Based on personal judgment or expertise

Probability Rules

• $0 \le P(A) \le 1$ for any event $A$
• $P(S) = 1$ (something must happen)
• $P(\emptyset) = 0$ (impossible event)

Basic Probability

For equally likely outcomes:

$P(A) = \frac{|A|}{|S|} = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

Addition Rule (OR)

For the probability that event $A$ or event $B$ occurs:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

If $A$ and $B$ are mutually exclusive (cannot happen together):

$P(A \cup B) = P(A) + P(B)$

Multiplication Rule (AND)

For the probability that event $A$ and event $B$ both occur:

$P(A \cap B) = P(A) \cdot P(B|A)$

If $A$ and $B$ are independent:

$P(A \cap B) = P(A) \cdot P(B)$

Conditional Probability

The probability of $A$ given that $B$ has occurred:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Binomial Probability

The probability of exactly $k$ successes in $n$ independent trials, each with probability $p$:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Summary Table

• Assuming events are independent when they are not — drawing cards without replacement changes probabilities after each draw.
• Forgetting to subtract the overlap in the addition rule — when events can occur together, you must subtract $P(A \cap B)$ to avoid double-counting.
• Confusing "and" with "or" — "and" means both events happen (multiply probabilities for independent events); "or" means at least one happens (add probabilities).
• Not considering all possible outcomes in the sample space — make sure to count the total correctly, especially with combinations and permutations.
• Confusing conditional probability direction — $P(A|B)$ is not the same as $P(B|A)$.