[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"tool-content:en:probability-calculator":3,"glossary:en":56,"published-tools-en":105},{"id":4,"documentId":5,"slug":6,"intro":7,"howTo":8,"longContent":9,"createdAt":10,"updatedAt":11,"publishedAt":12,"locale":13,"name":14,"faq":15,"examples":40,"category":41,"seo":50,"localizations":55,"metaTitle":52,"metaDescription":53},335,"h34dyn3un4ngghs2qssglqn5","probability-calculator","\u003Cp>Rolling a six on a fair die is a \u003Cstrong>1 in 6\u003C\u002Fstrong> chance, which this \u003Cstrong>probability calculator\u003C\u002Fstrong> shows at once as 0.166667, \u003Cstrong>16.67%\u003C\u002Fstrong> and odds of \u003Cstrong>1:5\u003C\u002Fstrong>. Three modes handle the three shapes the question takes, so a single event out of a set of outcomes sits alongside two independent events combined and the binomial case of \u003Cstrong>k successes in n trials\u003C\u002Fstrong>. The two-event mode multiplies for \"A and B\" then subtracts that overlap for \"A or B\", so its answers hold while the events stay independent.\u003C\u002Fp>","\u003Col>\u003Cli>Pick a mode. Single event counts favourable outcomes out of a total; Two events combines A and B; Binomial handles k successes in n repeated trials.\u003C\u002Fli>\u003Cli>Enter probabilities as decimals between 0 and 1. Type 0.25 for a 25% chance, as anything above 1 returns a range error.\u003C\u002Fli>\u003Cli>Read the headline percentage, then the decimal, the odds and the \"1 in N\" badge beside it. Every row of the results table carries all four.\u003C\u002Fli>\u003Cli>Open the formula panel to check which calculation ran, including the C(n, k) count in binomial mode. Trials stop at 1000.\u003C\u002Fli>\u003C\u002Fol>","\u003Ch2 id=\"how-to-calculate-probability\">How do you calculate probability?\u003C\u002Fh2>\n\u003Cp>A single probability is the count of favourable outcomes divided by the count of all possible outcomes. Take one die as an example: one face out of six gives the fraction 1\u002F6, a decimal of 0.166667, \u003Cstrong>16.67%\u003C\u002Fstrong> and exact odds of \u003Cstrong>1:5\u003C\u002Fstrong>. The complement, meaning every roll that is not a six, takes the remaining \u003Cstrong>83.33%\u003C\u002Fstrong>.\u003C\u002Fp>\n\u003Cp>That division holds only while every outcome carries the same weight, which covers a fair die or a shuffled deck. Drawing an ace from a standard 52-card deck counts 4 favourable cards out of 52, reduced to 1\u002F13, worth \u003Cstrong>7.69%\u003C\u002Fstrong> at odds of 1:12. A loaded die breaks the rule. Once one face turns up 30% of the time, counting faces gives the wrong answer and the measured 0.3 has to go straight into a probability field.\u003C\u002Fp>\n\u003Ch2 id=\"independent-mutually-exclusive\">Are two events independent or mutually exclusive?\u003C\u002Fh2>\n\u003Cp>The two-event mode assumes \u003Cstrong>independence\u003C\u002Fstrong> throughout, meaning one event happening leaves the odds of the other untouched. It multiplies for the joint case, written \u003Cstrong>P(A and B) = P(A) × P(B)\u003C\u002Fstrong>. For the union it adds the two probabilities then takes the overlap back out, written \u003Cstrong>P(A or B) = P(A) + P(B) − P(A) × P(B)\u003C\u002Fstrong>.\u003C\u002Fp>\n\u003Cp>With A at 0.5 and B at 0.25 the results table returns \u003Cstrong>12.5%\u003C\u002Fstrong> for both happening, \u003Cstrong>62.5%\u003C\u002Fstrong> for either one, 37.5% for neither and \u003Cstrong>50%\u003C\u002Fstrong> for exactly one of the two. At least one lands on 62.5% as well. It is the same question from the other side, 1 minus the 37.5% chance that neither shows up.\u003C\u002Fp>\n\u003Cp>Mutually exclusive events cannot both land on the same trial, which rewrites both formulas. Their joint probability is \u003Cstrong>0\u003C\u002Fstrong> and their union becomes a plain sum, \u003Cstrong>P(A or B) = P(A) + P(B)\u003C\u002Fstrong>. One card that is an ace or a king is the textbook case, worth 8 out of 52 at 15.38%.\u003C\u002Fp>\n\u003Cp>Feed those two 7.69% events into the two-event mode and the answer comes back at \u003Cstrong>14.79%\u003C\u002Fstrong>, short by 0.59 points, which is exactly the phantom overlap it subtracted. Staying in the single-event mode and entering \u003Cstrong>8 favourable out of 52\u003C\u002Fstrong> returns the true 15.38%, at the fraction 2\u002F13 and exact odds of 2:11.\u003C\u002Fp>\n\u003Ch2 id=\"binomial-probability\">What does the binomial probability calculator work out?\u003C\u002Fh2>\n\u003Cp>Ten flips of a coin can land three heads in \u003Cstrong>C(10, 3) = 120\u003C\u002Fstrong> different orders, which the binomial mode turns into an exact probability of \u003Cstrong>11.72%\u003C\u002Fstrong>, a cumulative \u003Cstrong>17.19%\u003C\u002Fstrong> for at most three heads and \u003Cstrong>94.53%\u003C\u002Fstrong> for at least three. The mean is \u003Cstrong>5 heads\u003C\u002Fstrong> with a standard deviation of \u003Cstrong>1.58\u003C\u002Fstrong>, so three heads falls 1.3 deviations below the middle of the binomial distribution.\u003C\u002Fp>\n\u003Cp>A player shooting \u003Cstrong>80%\u003C\u002Fstrong> from the free-throw line hits exactly 4 of 5 attempts on \u003Cstrong>40.96%\u003C\u002Fstrong> of trips, while 4 or more comes up \u003Cstrong>73.73%\u003C\u002Fstrong> of the time. Any run of repeated attempts at a fixed probability with two outcomes each fits the same mode, including a sampling check on a production batch. The calculator accepts up to \u003Cstrong>1000 trials\u003C\u002Fstrong>, beyond which the binomial coefficients outgrow floating-point precision.\u003C\u002Fp>\n\u003Ch2 id=\"theoretical-experimental\">How do you calculate theoretical and experimental probability?\u003C\u002Fh2>\n\u003Cp>A fair coin has a \u003Cstrong>50%\u003C\u002Fstrong> chance of heads on any single flip, settled by counting two faces without tossing anything. That counting on paper is \u003Cstrong>theoretical probability\u003C\u002Fstrong>, the job all three modes do. \u003Cstrong>Experimental probability\u003C\u002Fstrong>, also called empirical probability, comes from what somebody observed, dividing recorded successes by recorded trials.\u003C\u002Fp>\n\u003Cp>Both use the same division, though the two numbers answer different questions. Flip a coin 100 times and log 47 heads and the experimental probability is \u003Cstrong>47%\u003C\u002Fstrong>, which the single-event mode returns from 47 favourable out of 100. That gap against the theoretical 50% narrows as the trial count grows, an effect known as the law of large numbers.\u003C\u002Fp>\n\u003Ch2 id=\"four-forms\">Why show a probability four different ways?\u003C\u002Fh2>\n\u003Cp>Each row of the results table shows a decimal, a percentage, odds and a 1 in N reading, since each one suits a different setting. Statistics work runs on the decimal, a report reads better in percent while bookmakers and medical risk sheets quote odds. The 1 in N form is the one most people picture, so 0.1171875 becomes \u003Cstrong>1 chance in 8.53\u003C\u002Fstrong>. Every figure prints to six decimals, so the 16.67% above reads 16.666667 on screen.\u003C\u002Fp>\n\u003Ctable>\n\u003Cthead>\u003Ctr>\u003Cth>Single event\u003C\u002Fth>\u003Cth>Fraction\u003C\u002Fth>\u003Cth>Percent\u003C\u002Fth>\u003Cth>Exact odds\u003C\u002Fth>\u003Cth>1 in N\u003C\u002Fth>\u003C\u002Ftr>\u003C\u002Fthead>\n\u003Ctbody>\n\u003Ctr>\u003Ctd>Heads on one coin flip\u003C\u002Ftd>\u003Ctd>1\u002F2\u003C\u002Ftd>\u003Ctd>50%\u003C\u002Ftd>\u003Ctd>1:1\u003C\u002Ftd>\u003Ctd>2\u003C\u002Ftd>\u003C\u002Ftr>\n\u003Ctr>\u003Ctd>A six on one die\u003C\u002Ftd>\u003Ctd>1\u002F6\u003C\u002Ftd>\u003Ctd>16.67%\u003C\u002Ftd>\u003Ctd>1:5\u003C\u002Ftd>\u003Ctd>6\u003C\u002Ftd>\u003C\u002Ftr>\n\u003Ctr>\u003Ctd>A spade from a 52-card deck\u003C\u002Ftd>\u003Ctd>1\u002F4\u003C\u002Ftd>\u003Ctd>25%\u003C\u002Ftd>\u003Ctd>1:3\u003C\u002Ftd>\u003Ctd>4\u003C\u002Ftd>\u003C\u002Ftr>\n\u003Ctr>\u003Ctd>An ace or a king in one draw\u003C\u002Ftd>\u003Ctd>2\u002F13\u003C\u002Ftd>\u003Ctd>15.38%\u003C\u002Ftd>\u003Ctd>2:11\u003C\u002Ftd>\u003Ctd>6.5\u003C\u002Ftd>\u003C\u002Ftr>\n\u003Ctr>\u003Ctd>An ace from a 52-card deck\u003C\u002Ftd>\u003Ctd>1\u002F13\u003C\u002Ftd>\u003Ctd>7.69%\u003C\u002Ftd>\u003Ctd>1:12\u003C\u002Ftd>\u003Ctd>13\u003C\u002Ftd>\u003C\u002Ftr>\n\u003C\u002Ftbody>\n\u003C\u002Ftable>\n\u003Cp>Odds of \u003Cstrong>1:12\u003C\u002Fstrong> on that ace count 4 favourable cards against the other 48, reduced to one against twelve. The probability is 1\u002F13, while 1\u002F12 would be 8.33%. Taking the second number of an odds pair for the total is the usual slip.\u003C\u002Fp>","2026-08-06T07:18:52.663Z","2026-08-25T10:00:01.949Z","2026-08-25T10:00:00.000Z","en","Probability Calculator",[16,20,24,28,32,36],{"id":17,"question":18,"answer":19},2361,"What is the formula for probability?","\u003Cp>The three modes run on three formulas.\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>Single event\u003C\u002Fstrong> divides favourable outcomes by total outcomes. One face of a die gives 1 ÷ 6 = \u003Cstrong>16.67%\u003C\u002Fstrong>.\u003C\u002Fli>\u003Cli>\u003Cstrong>Two independent events\u003C\u002Fstrong> multiply for the joint case as P(A and B) = P(A) × P(B). The union follows as P(A or B) = P(A) + P(B) − P(A) × P(B).\u003C\u002Fli>\u003Cli>\u003Cstrong>Binomial\u003C\u002Fstrong> uses P(X = k) = C(n, k) × p\u003Csup>k\u003C\u002Fsup> × (1 − p)\u003Csup>n − k\u003C\u002Fsup>. The formula panel prints it with the numbers filled in.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>The complement rule sits underneath all three and gives P(not A) = 1 − P(A).\u003C\u002Fp>",{"id":21,"question":22,"answer":23},2362,"What is the probability of rolling a 6 on a die?","\u003Cp>One favourable face out of six equally likely faces makes \u003Cstrong>1\u002F6\u003C\u002Fstrong>, which the calculator writes as 0.166667, \u003Cstrong>16.67%\u003C\u002Fstrong>, exact odds of \u003Cstrong>1:5\u003C\u002Fstrong> and 1 chance in 6. The complement row covers the other five faces at \u003Cstrong>83.33%\u003C\u002Fstrong>. Two dice both landing on six is a different question that belongs in the two-event mode at 1\u002F6 each, worth \u003Cstrong>2.78%\u003C\u002Fstrong> or 1 chance in 36.\u003C\u002Fp>",{"id":25,"question":26,"answer":27},2363,"How do you calculate the probability of two events happening?","\u003Cp>Multiply the two probabilities when the events are independent. Enter 0.5 and 0.25 in the two-event mode and the \"A and B\" row returns \u003Cstrong>12.5%\u003C\u002Fstrong>, one chance in 8. Two situations break that multiplication.\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>Mutually exclusive events\u003C\u002Fstrong> never happen together, which makes P(A and B) = 0. A single card cannot be an ace and a king at once.\u003C\u002Fli>\u003Cli>\u003Cstrong>Dependent events\u003C\u002Fstrong> need the conditional probability in the second slot. Drawing two aces from a deck runs 4\u002F52 then 3\u002F51, entered as 0.0769 and 0.0588 for a result of \u003Cstrong>0.45%\u003C\u002Fstrong>, about 1 chance in 221.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>In that dependent case only the \"A and B\" row stays valid, as every other row keeps assuming independence.\u003C\u002Fp>",{"id":29,"question":30,"answer":31},2364,"What does \"at least one\" mean in probability?","\u003Cp>At least one means one occurrence or more, which covers everything except neither event happening. The calculator works it out as \u003Cstrong>1 − P(neither)\u003C\u002Fstrong>, so events at 0.5 and 0.25 give 1 − 0.375 = \u003Cstrong>62.5%\u003C\u002Fstrong>. That figure matches the \"A or B\" row exactly, since both describe the same set of outcomes. \"Exactly one\" is the narrower question, worth \u003Cstrong>50%\u003C\u002Fstrong> on the same pair.\u003C\u002Fp>",{"id":33,"question":34,"answer":35},2365,"What is the difference between theoretical and experimental probability?","\u003Cp>Theoretical probability counts what should happen from the structure of the situation, such as \u003Cstrong>50%\u003C\u002Fstrong> heads on a fair coin. Experimental probability records what did happen, dividing observed successes by observed trials, which turns 47 heads in 100 flips into \u003Cstrong>47%\u003C\u002Fstrong>. Both go into the single-event mode as favourable out of total. The longer the run of trials, the closer the experimental figure settles towards the theoretical one.\u003C\u002Fp>",{"id":37,"question":38,"answer":39},2366,"How do you read odds of 1:5?","\u003Cp>Odds of \u003Cstrong>1:5\u003C\u002Fstrong> put one favourable outcome against five unfavourable ones, which makes six cases in total and a probability of \u003Cstrong>1\u002F6\u003C\u002Fstrong>, about 16.67%. Odds count the two sides against each other while a probability counts one side against the whole. Reading 1:5 as one chance in five would push a 16.67% event up to 20%.\u003C\u002Fp>",[],{"id":42,"documentId":43,"uid":44,"name":45,"tagline":46,"hubContent":47,"createdAt":48,"updatedAt":48,"publishedAt":49,"locale":13},15,"s8cujbpmiszotf6zdbotc2p0","math","Math","Calculators for school, work and everyday numbers","\u003Cp>Every calculator in this category computes live as you type and shows the formula behind the result. Grades, percentages, fractions, ratios or volumes: you see the answer and the reasoning, so you can trust the number you copy. Each tool also documents its edge cases, because a calculator you cannot verify is just a guess with confidence.\u003C\u002Fp>","2026-07-17T11:46:54.883Z","2026-07-17T12:01:48.544Z",{"id":51,"metaTitle":52,"metaDescription":53,"keywords":54,"metaRobots":54,"structuredData":54,"metaViewport":54,"canonicalURL":54},568,"Probability Calculator: Single, Two Events and Binomial","Probability calculator for one event, two independent events or the binomial case of k successes in n trials. Decimal, percent, odds and 1 in N.",null,[],[57,69,81,93],{"id":58,"documentId":59,"slug":60,"term":61,"definition":62,"relatedTools":63,"createdAt":66,"updatedAt":67,"publishedAt":68,"locale":13},17,"o1ljpluv6app6z9rrhst4gfn","cubic-yard","Cubic yard","\u003Cp>A \u003Cstrong>cubic yard\u003C\u002Fstrong> is the volume of a cube measuring 3 x 3 x 3 feet, which equals 27 cubic feet or about \u003Cstrong>0.7646 m³\u003C\u002Fstrong>. It is the standard unit for ordering bulk materials in the United States: ready-mix concrete, topsoil, gravel, sand and mulch are all priced and delivered by the cubic yard, often shortened to \"yard\" on a quote.\u003C\u002Fp>\u003Cp>For scale, a typical 12 x 10 ft patio slab poured 4 inches thick contains 40 cubic feet, which is 40 \u002F 27 = 1.48 cubic yards of concrete. A standard ready-mix truck carries 8 to 10 cubic yards, so that whole slab uses less than a fifth of one load.\u003C\u002Fp>\u003Cp>How big is a cubic yard in practice? The cube stands 3 feet on each side, roughly counter height, and holds about 14 full wheelbarrow loads of material. Spread out, one cubic yard covers close to 100 square feet at a depth of 3 inches, the usual layer for mulch or a gravel path.\u003C\u002Fp>",[64,65],"math\u002Fconcrete-calculator","math\u002Fvolume-calculator","2026-07-18T14:23:56.812Z","2026-08-02T10:00:01.228Z","2026-08-02T10:00:00.000Z",{"id":70,"documentId":71,"slug":72,"term":73,"definition":74,"relatedTools":75,"createdAt":78,"updatedAt":79,"publishedAt":80,"locale":13},16,"mdw0ypy9f3z1nioxaxnl9rvr","password-entropy","Entropy (passwords)","\u003Cp>\u003Cstrong>Entropy\u003C\u002Fstrong> measures how unpredictable a password is, expressed in bits. Each additional bit doubles the number of guesses an attacker needs. For a randomly generated password, it follows a simple formula: entropy = length × log2 of the alphabet size, so both the length and the variety of characters raise the score.\u003C\u002Fp>\u003Cp>A random 16-character password drawn from the 94 printable ASCII symbols reaches 16 × log2(94), about \u003Cstrong>105 bits\u003C\u002Fstrong>. At 10 billion guesses per second, exhausting that space would take on the order of 10^14 years, while a 6-character password from the same alphabet peaks at 39 bits and falls in about a minute.\u003C\u002Fp>\u003Cp>Written as a formula, \u003Cstrong>E = L × log2(N)\u003C\u002Fstrong>: the entropy equals the length L multiplied by the base-2 logarithm of N, the number of possible symbols per character. Current guidance treats 75 to 80 bits as the comfortable minimum for accounts that matter, a level a random 12-character password mixing all character types already clears.\u003C\u002Fp>",[76,77],"generator\u002Fpassword-generator","generator\u002Fusername-generator","2026-07-18T14:23:55.755Z","2026-08-01T10:00:01.360Z","2026-08-01T10:00:00.000Z",{"id":42,"documentId":82,"slug":83,"term":84,"definition":85,"relatedTools":86,"createdAt":90,"updatedAt":91,"publishedAt":92,"locale":13},"ypkfs6wnfljm03ge2b1ztiwv","percentage-point","Percentage point","\u003Cp>A \u003Cstrong>percentage point\u003C\u002Fstrong> is the unit used to express the arithmetic difference between two percentages. It compares rates by subtraction, while \"percent\" compares them by division: the two measures answer different questions and can differ wildly for the same change.\u003C\u002Fp>\u003Cp>Example: an interest rate that moves from 5% to 7% rises by \u003Cstrong>2 percentage points\u003C\u002Fstrong> (7 - 5 = 2), but by 40 percent in relative terms (2 ÷ 5 × 100 = 40). Saying \"the rate went up 2%\" would be wrong on both counts: the correct phrasings are \"up 2 points\" or \"up 40%\".\u003C\u002Fp>\u003Cp>The standard abbreviation is \u003Cstrong>pp\u003C\u002Fstrong>, sometimes just \"points\", as in \"unemployment fell by 0.5 pp\". Confusing percent with percentage points remains the classic trap: a party moving from 20% to 30% in the polls gains 10 percentage points, yet grows by 50 percent, and headlines regularly pick the wrong figure.\u003C\u002Fp>",[87,88,89],"math\u002Fpercentage-calculator","math\u002Faverage-calculator","finance\u002Fcalcul-remise","2026-07-18T14:23:54.573Z","2026-07-31T10:00:01.940Z","2026-07-31T10:00:00.000Z",{"id":94,"documentId":95,"slug":96,"term":97,"definition":98,"relatedTools":99,"createdAt":102,"updatedAt":103,"publishedAt":104,"locale":13},14,"g3okb797ekrz0kkpm36mfjwc","weighted-average","Weighted average","\u003Cp>A \u003Cstrong>weighted average\u003C\u002Fstrong> is an average where each value counts in proportion to an assigned importance, called its weight, instead of counting equally. Each value is multiplied by its weight, the products are added, and the sum is divided by the total of the weights. Class grades, GPAs, and stock indexes are all weighted averages.\u003C\u002Fp>\u003Cp>Example: scores of 92, 85, and 88 with weights of 30, 30, and 40 give (92 x 30 + 85 x 30 + 88 x 40) \u002F 100 = \u003Cstrong>88.3\u003C\u002Fstrong>. The plain average of the same scores is 88.33, but if the heaviest score dropped to 78, the weighted average would fall to 84.3 while the plain average only fell to 85.\u003C\u002Fp>\u003Cp>The weighted average formula in plain words: multiply each value by its weight, add up the products, then divide by the sum of the weights. In a spreadsheet the whole calculation fits in one cell, since Excel's SUMPRODUCT function multiplies and adds in a single pass: SUMPRODUCT of the values and weights, divided by SUM of the weights.\u003C\u002Fp>",[88,100,101],"math\u002Fgrade-calculator","math\u002Fgpa-calculator","2026-07-18T14:23:53.329Z","2026-07-30T10:00:01.216Z","2026-07-30T10:00:00.000Z",{"slugs":106},[107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,6,157,158,159,160,161],"age-calculator","average-calculator","cd-calculator","concrete-calculator","cursive-font-generator","date-calculator","fantasy-name-generator","final-grade-calculator","fraction-calculator","glitch-text-generator","gpa-calculator","grade-calculator","hex-converter","hours-calculator","interest-calculator","military-time-converter","roman-numeral-converter","password-generator","binary-converter","celsius-to-fahrenheit-converter","kg-to-lbs-converter","percentage-calculator","word-counter","tip-calculator","morse-code-translator","username-generator","ratio-calculator","sales-tax-calculator","small-text-generator","mm-to-inches-converter","volume-calculator","timer","square-footage-calculator","stair-calculator","roof-pitch-calculator","board-foot-calculator","aspect-ratio-calculator","rounding-calculator","random-letter-generator","mulch-calculator","fence-calculator","upside-down-text-generator","acreage-calculator","gravel-calculator","proportion-calculator","paint-calculator","tile-calculator","braille-translator","quadratic-formula-calculator","gcf-calculator","hex-to-rgb-converter","uuid-generator","standard-deviation-calculator","lcm-calculator","caesar-cipher"]