Probability Calculator

Probability is the count of outcomes you want divided by the count of outcomes possible, assuming every outcome is equally likely.

P(A) = Favorable Outcomes / Total Outcomes

Drawing an ace from a standard deck: 4 aces out of 52 cards, so P = 4/52 = 0.0769, or 7.69%. The answer is always between 0 (impossible) and 1 (certain), and the two are exhaustive: whatever probability is not assigned to the event belongs to its complement.

P(not A) = 1 − P(A)

Probability is not odds, and the difference matters

This trips people up constantly, particularly anyone reading betting markets. Probability compares favorable outcomes to the total. Odds compare favorable outcomes to unfavorable ones.

For the ace draw: the probability is 4/52, but the odds in favor are 4:48, which simplifies to 1:12. Odds against are 48:4, or 12:1. A bookmaker quoting “12 to 1” is describing the same event as “7.69%”, not a different one. To convert: probability = favorable / (favorable + unfavorable), and odds = probability / (1 − probability).

Repeated trials, and why “at least once” grows fast

Given one chance in n, people badly underestimate how quickly repeated attempts accumulate. The chance of the event happening at least once in k independent trials is not k × P. That would exceed 1 eventually, which is nonsense. It is the complement of never happening at all:

P(at least one in k trials) = 1 − (1 − P)^k

With a 7.69% single-draw chance, five draws with replacement gives 1 − 0.9231⁵ = 32.98%, not 38.5%. The gap widens with more trials. A useful rule of thumb: it takes roughly 0.69 / P trials to pass a 50% chance, so a 1-in-100 event needs about 69 attempts to become more likely than not, not 50.

Expected value

Expected value is the average outcome per trial over the long run, found by weighting each payout by its probability:

EV = P(win) × payout

At 7.69% with a $100 payout, EV is $7.69 per trial. That is what the trial is worth before you subtract what it costs to enter. A trial priced above its expected value loses money on average no matter how the individual attempts land, which is the entire business model of every lottery ever run.

Expected value says nothing about any single trial. You will never actually receive $7.69; you receive $100 or nothing. It only describes the average across many repetitions, which is why it is a poor guide for one-off decisions with outcomes you cannot afford.

Combining events

For independent events, where one has no effect on the other: P(A and B) = P(A) × P(B). Two aces drawn with replacement: 0.0769 × 0.0769 = 0.59%.

For mutually exclusive events, which cannot both happen: P(A or B) = P(A) + P(B). Drawing an ace or a king: 4/52 + 4/52 = 15.4%.

If the events can overlap, subtract the double-counted part: P(A or B) = P(A) + P(B) − P(A and B). Drawing an ace or a spade is 4/52 + 13/52 − 1/52 = 30.8%, because the ace of spades would otherwise be counted twice.