Find remainder when dividing a polynomial by x - a using P(a). Enter polynomial and a; see steps, checks, explanation.
x, ^ for powers, and * for multiplication (optional).
Examples: 2*x^3 + 3*x - 5, x^4 - x + 1.
P(a).
If the remainder is 0, then (x - a) is a factor of P(x).
Remainder when dividing P(x) by (x - a) is P(a). Example: P(2) for P(x)=x^3-4x^2+x+6 → 0.
P(x) and number a. The remainder is P(a). :contentReference[oaicite:1]{index=1}^ (e.g., x^4) and multiply as 2*x if needed.P(a)=0, then (x-a) is a factor (Factor Theorem). :contentReference[oaicite:2]{index=2}A Conditional Probability Calculator evaluates probabilities that depend on given information, such as “the probability of event \(A\) occurring given that \(B\) has occurred,” written \(P(A\mid B)\). It simplifies everyday reasoning—medical tests, spam filtering, reliability, quality control—by turning evidence into updated beliefs. The tool accepts raw counts (contingency tables), named partitions, or direct probabilities, and then computes \(P(A\mid B)\), \(P(B\mid A)\), joint probabilities \(P(A\cap B)\), complements, and independence checks. It also implements Bayes’ theorem to invert conditionals when direct measurement is hard, providing transparent steps and an equation preview for clear interpretation.
Conditional probability scales the likelihood of \(A\) by restricting to the context where \(B\) is true. In datasets, this means focusing on the subset matching \(B\). The calculator supports three common workflows: (1) Counts—enter a \(2\times2\) or larger table and it derives all proportions; (2) Direct probabilities—enter \(P(A)\), \(P(B)\), and either \(P(A\cap B)\) or \(P(B\mid A)\) to fill the rest; (3) Partitions—enter priors \(P(A_i)\) and likelihoods \(P(B\mid A_i)\) to compute \(P(B)\) and posteriors \(P(A_i\mid B)\). The tool flags impossible inputs (e.g., probabilities outside \([0,1]\), or \(P(A\cap B)>P(B)\)) and notes when \(P(B)=0\) makes \(P(A\mid B)\) undefined.
Definition: \[ P(A\mid B)=\frac{P(A\cap B)}{P(B)},\quad P(B)>0. \]
Bayes’ theorem (single hypothesis): \[ P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B)}. \]
Law of total probability: \[ P(B)=\sum_{i} P(B\mid A_i)\,P(A_i),\quad \{A_i\}\ \text{a partition}. \]
Bayes (multiple hypotheses): \[ P(A_k\mid B)=\frac{P(B\mid A_k)\,P(A_k)}{\sum_i P(B\mid A_i)\,P(A_i)}. \]
Independence test: \[ A\ \text{independent of}\ B \iff P(A\mid B)=P(A) \iff P(A\cap B)=P(A)P(B). \]
Disease prevalence \(P(D)=0.02\); sensitivity \(P(+\mid D)=0.95\); false positive rate \(P(+\mid \overline{D})=0.05\). \(P(+)=0.95(0.02)+0.05(0.98)=0.068\). Posterior \(P(D\mid +)=\dfrac{0.95\cdot0.02}{0.068}\approx0.279\) (about 27.9%).
From a standard deck, \(A=\) “red,” \(B=\) “face card.” \(P(B)=12/52\), \(P(A\cap B)=6/52\). \(P(A\mid B)=\dfrac{6/52}{12/52}=1/2\).
Spam filter: 300 spam (of which 270 flagged “X”), 700 ham (70 flagged “X”). \(P(X\mid \text{spam})=270/300=0.90\), \(P(X\mid \text{ham})=70/700=0.10\), \(P(\text{spam})=0.3\). \(P(X)=0.9\cdot0.3+0.1\cdot0.7=0.34\). \(P(\text{spam}\mid X)=\dfrac{0.9\cdot0.3}{0.34}\approx0.794\).
It’s the probability of an event given that another event is known to have occurred.
Use it to invert a conditional (from \(P(B\mid A)\) to \(P(A\mid B)\)) when you know priors and likelihoods.
If \(P(A\mid B)=P(A)\) (or \(P(A\cap B)=P(A)P(B)\)), the events are independent.
Then \(P(A\mid B)\) is undefined; there’s no sample space where \(B\) occurs to condition on.
Yes. Convert counts to proportions by dividing by totals; the calculator does this automatically.
Rare conditions (low prior) with nonzero false-positive rates can yield modest \(P(D\mid +)\) despite high sensitivity.