Basic probability
P(E) = favorable outcomes / total outcomes
Use when the outcomes are equally likely and you can count successes directly from the full sample space.
Example use: Pull a specific card from a deck in one draw.
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Probability formulas reference
A short stable reference for the main probability formulas used across the calculators, answers, and problems on this site.
Complement rule
P(not E) = 1 - P(E)
Use when the opposite event is easier to count, especially for phrases like at least one.
Example use: Find the probability of at least one head in 4 coin flips by subtracting the probability of zero heads.
Binomial exactly
P(X = k) = C(n, k) p^k (1 - p)^(n - k)
Use for repeated independent trials with the same success probability, such as coin flips.
Example use: Get exactly 3 heads in 5 fair coin flips.
Binomial range
P(X at least or at most k) = sum of exact binomial terms
Use when the wording asks for a range of success counts rather than one exact value.
Example use: Get at least 2 heads in 4 fair coin flips.
At least one shortcut
P(at least 1 success) = 1 - P(0 successes)
Use when the complement is shorter than summing every successful case one by one.
Example use: Find the chance of drawing at least 1 ace in a 5-card hand.
Hypergeometric exactly
P(X = k) = C(K, k) C(N - K, n - k) / C(N, n)
Use for drawing without replacement from a finite population, such as cards from a deck.
Example use: Draw exactly 2 aces in a 5-card hand.
Hypergeometric range
P(X at least or at most k) = sum of exact hypergeometric terms
Use when without-replacement wording asks for a range of possible success counts.
Example use: Draw at most 1 ace in a 5-card hand.
Model choice
Same p each trial -> binomial. No replacement -> hypergeometric.
Use binomial when trials stay independent with the same success chance. Use hypergeometric when each draw changes what remains.
Example use: Use binomial for 5 coin flips, but use hypergeometric for a 5-card hand from a deck.