Probability Calculator | Events

These two concepts are often confused but are fundamentally different:

• ›Mutually exclusive (disjoint), A and B cannot both happen. P(A ∩ B) = 0. The addition rule simplifies to P(A ∪ B) = P(A) + P(B). Example: a single die roll is ≤3 OR ≥4, these outcomes cannot overlap.
• ›Independent, knowing A occurred gives no information about B. P(A|B) = P(A), equivalently P(A ∩ B) = P(A) × P(B). Example: flipping a coin does not affect the next flip.
• ›The key difference, mutually exclusive events are always dependent (if A happened, B definitely didn't). Independent events can (and usually do) overlap.
• ›Both at once, two events can be independent but not mutually exclusive (most common), mutually exclusive but not independent (rare), or neither (general case).

Conditional probability is one of the most powerful and widely used concepts in statistics:

• ›Definition, P(A|B) = P(A ∩ B) / P(B). The probability space is restricted to B having occurred, so we scale by P(B) to renormalize.
• ›Medical diagnostics, sensitivity = P(positive test | disease). Specificity = P(negative test | no disease). PPV = P(disease | positive test), requires Bayes' theorem.
• ›Bayes' theorem, P(A|B) = P(B|A) × P(A) / P(B). Lets you update a prior belief (P(A)) given new evidence (P(B|A)) to form a posterior (P(A|B)).
• ›Spam filters, P(spam | word "prize") is estimated from training data. The Naive Bayes classifier multiplies conditional probabilities across all words to classify messages.

The complement rule is one of the most practically useful shortcuts in probability:

• ›Formula, P(A′) = 1 − P(A). Follows from the axiom that the total probability of all outcomes is 1.
• ›"At least one" problems, P(at least one success) = 1 − P(zero successes). Often the right side is a simple product of failure probabilities.
• ›Birthday problem, P(at least two people share a birthday in a room of 23) = 1 − P(all birthdays distinct) = 1 − (365×364×⋯×343)/365²³ ≈ 50.7%. Surprises almost everyone.
• ›Reliability engineering, P(system fails) = 1 − P(system works). P(system works) = product of component reliabilities for series systems.

Odds and probability are two different ways to express the same underlying chance:

• ›Odds in favor, favorable : unfavorable. P = 0.25 → odds 1:3 (win once for every three losses).
• ›Odds against, unfavorable : favorable. P = 0.25 → odds against 3:1.
• ›Converting probability to odds, odds in favor = P/(1−P). P = 0.8 → odds = 4:1.
• ›Converting odds to probability, P = a/(a+b) where odds are a:b. Odds 5:2 → P = 5/7 ≈ 0.714.
• ›Sports betting, decimal odds of 2.50 mean P ≈ 1/2.50 = 40%. The bookmaker's overround (all implied probabilities sum > 100%) is the house edge.

The addition rule handles the "or" case in probability:

• ›General formula, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Without subtracting the intersection, overlapping outcomes would be counted twice.
• ›Mutually exclusive case, if A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B). Coin shows heads or tails: P(H ∪ T) = 0.5 + 0.5 = 1.
• ›Three events, P(A∪B∪C) = P(A)+P(B)+P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). The pattern is inclusion-exclusion.
• ›Venn diagram, think of A and B as overlapping circles. Addition counts the union; subtracting the intersection removes the double-counted middle region.

Bayes' theorem is arguably the most important formula in applied probability:

• ›Formula, P(A|B) = P(B|A) × P(A) / P(B). P(A) is the prior; P(A|B) is the posterior (updated belief after evidence B).
• ›Medical testing, P(disease|+) = P(+|disease) × P(disease) / P(+). Even with a 99% accurate test, if disease prevalence is 0.1%, most positives are false positives.
• ›Spam filtering, P(spam|words) = ∏ P(word|spam) × P(spam) / P(words). Naive Bayes assumes independence of words given class.
• ›Law of total probability, P(B) = P(B|A)P(A) + P(B|A′)P(A′). Often needed to compute the denominator in Bayes' theorem.

Interpreting probabilities in practical contexts requires care:

• ›Frequentist interpretation, P = 0.05 means in a long run of identical experiments, the event occurs 5% of the time. In 1,000 coin tosses, about 50 would land as "heads twice in a row".
• ›p-value in statistics, a p-value ≤ 0.05 means the observed data would occur by chance less than 5% of the time under the null hypothesis. It does NOT mean there is a 95% chance the hypothesis is false.
• ›Rare events, a 1% probability event is expected once in 100 trials. Over 100 trials, P(never occurring) = (0.99)¹⁰⁰ ≈ 36.6%, so seeing it not occur is still reasonably likely.
• ›Risk communication, "1 in 1,000 chance of side effects" = P = 0.001. In a drug prescribed to 10 million people, about 10,000 would experience side effects even with this low probability.