30:[] 2e:{"title":"Bayes Theorem","cells":[{"id":0,"value":"md`# Bayes Theorem`","pinned":false,"hidden":false,"mode":"ojs"},{"id":22,"value":"svg = {\n const svg = html``\n const container = d3.select(svg)\n \n const outer = container.append(\"g\").attr(\"transform\",`translate(300,150)`)\n \n outer.append(\"circle\")\n .attr(\"r\",100)\n .attr(\"stroke\",\"black\")\n .attr(\"fill\",\"skyblue\")\n .attr(\"cx\",-50)\n .attr(\"opacity\",.5)\n \n \n outer.append(\"circle\")\n .attr(\"r\",100)\n .attr(\"stroke\",\"black\")\n .attr(\"fill\",\"tomato\")\n .attr(\"cx\",50)\n .attr(\"opacity\",.5)\n \n outer.append(\"text\")\n .text(\"A\")\n .attr(\"text-anchor\",\"middle\")\n .attr(\"x\",-50)\n .attr(\"y\",-110)\n \n outer.append(\"text\")\n .text(\"B\")\n .attr(\"text-anchor\",\"middle\")\n .attr(\"x\",50)\n .attr(\"y\",-110)\n \n \n \n return svg\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":49,"value":"html`

Conditional Probability Formula

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":52,"value":"tex`P(A|B) = \\dfrac{P(A \\cap B)}{P(B)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":57,"value":"html`The probability that Mahesh comes to the party tonight is 0.7. The probability that both Mahesh and Chris are there is 0.4. What is the probability that Chris is there, given that Mahesh is there? `","pinned":false,"hidden":false,"mode":"ojs"},{"id":72,"value":"html`

Another Form

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":75,"value":"tex`P(B)P(A|B) = P(A \\cap B)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":88,"value":"tex`P(A)P(B|A) = P(A \\cap B)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":80,"value":"html`You have four marbles in a jar: red, blue, yellow, and white. You reach into the jar and draw out a marble. Then you reach into the jar and draw out a second marble. What is the probability you draw the red marble first followed by the blue marble?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"html`

What If?

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":99,"value":"tex`P(A|B) = P(A)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":102,"value":"tex`P(A \\cap B) = P(B)P(A|B) = P(B)P(A)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":110,"value":"tex`P(A \\cap B) = P(A)P(B)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":113,"value":"tex`\\text{If } P(A|B) = P(A), \\text{does } P(B|A) = P(B)?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"tex`P(B|A) = \\dfrac{P(A \\cap B)}{P(A)} = \\dfrac{P(A)P(B)}{P(A)} = P(B)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":121,"value":"html`

Independent Events Problem

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":124,"value":"html`You roll a blue die and a red die. What is the probability you roll a 3 with the blue die and a 7 with the red die?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":131,"value":"html`

Bayes' Theorem

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"tex`P(A|B) = \\dfrac{P(A \\cap B)}{P(B)} = \\dfrac{P(A)P(B|A)}{P(B)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":140,"value":"tex`P(A|B) = \\dfrac{P(A)P(B|A)}{P(B)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":145,"value":"html`

Bayes' Theorem Exercises

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":157,"value":"html`90% of those who failed an exam were from Redmond's class. 70% of those who took the test were from Redmond's class. 5% of students, overall, failed the test. What is the probability that a student failed given they were from Redmond's class?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":162,"value":"tex`A = \\text{Redmond } B = \\text{failed } P(A) = .7 \\text{ } P(B) = .05 \\text{ } P(A|B) = .90`","pinned":false,"hidden":false,"mode":"ojs"},{"id":170,"value":"tex`P(B|A) = \\dfrac{P(B)P(A|B)}{P(A)} = \\dfrac{.05*.9}{.7} = ${.05*.9/.7}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":175,"value":"html`Suppose that 2% of the population has a certain disease. You've developed a test that can detect this disease in a patient. 95% of the patients who have this disease will test positive. 2.5% of patients test positive. What is the probability a patient who tests positive has the disease?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":188,"value":"tex`A = disease`","pinned":false,"hidden":false,"mode":"ojs"},{"id":196,"value":"tex`B = positive`","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"tex`P(A) = .02`","pinned":false,"hidden":false,"mode":"ojs"},{"id":200,"value":"tex`P(B) = .025`","pinned":false,"hidden":false,"mode":"ojs"},{"id":202,"value":"tex`P(B|A) = .95`","pinned":false,"hidden":false,"mode":"ojs"},{"id":204,"value":"tex`P(A|B) = .02*.95/.025 = ${.02*.95/.025}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":219,"value":"html`

Another Helpful Formula

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":214,"value":"svg2 = {\n const svg = html``\n const container = d3.select(svg)\n \n const outer = container.append(\"g\").attr(\"transform\",`translate(300,150)`)\n \n outer.append(\"circle\")\n .attr(\"r\",100)\n .attr(\"stroke\",\"black\")\n .attr(\"fill\",\"skyblue\")\n .attr(\"cx\",-50)\n .attr(\"opacity\",.5)\n \n \n outer.append(\"circle\")\n .attr(\"r\",100)\n .attr(\"stroke\",\"black\")\n .attr(\"fill\",\"tomato\")\n .attr(\"cx\",50)\n .attr(\"opacity\",.5)\n \n outer.append(\"text\")\n .text(\"A\")\n .attr(\"text-anchor\",\"middle\")\n .attr(\"x\",-50)\n .attr(\"y\",-110)\n \n outer.append(\"text\")\n .text(\"B\")\n .attr(\"text-anchor\",\"middle\")\n .attr(\"x\",50)\n .attr(\"y\",-110)\n \n \n \n return svg\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":221,"value":"tex`P(\\text{not } A)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":228,"value":"tex`P(B) = P(A \\cap B) + P(\\text{not } A \\cap B)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":231,"value":"tex`P(B) = P(A)P(B|A) + P(\\text{not } A)P(B|\\text{not } A)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":234,"value":"tex`P(A|B) = \\dfrac{P(A)P(B|A)}{P(B)} = \\dfrac{P(A)P(B|A)}{P(A)P(B|A) + P(\\text{not } A)P(B|\\text{not } A)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":239,"value":"html`

The Phillies have made the World Series. The other team to play in the World Series will be determined tonight by a game between the A's and the Yankees. The Yankees have a 80% chance of winning that game.

If the Phillies face the Yankees, they have a 30% chance of winning the series. If they face the A's, they have a 60% chance of winning the series. What is the probability they faced the Yankees given that the Phillies won the World Series?

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":250,"value":"tex`A = \\text{Yankees make series}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":251,"value":"tex`B = \\text{Phillies win series}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":253,"value":"tex`P(A|B) = ?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":262,"value":"tex`P(A) = .8`","pinned":false,"hidden":false,"mode":"ojs"},{"id":263,"value":"tex`P(B|A) = .3`","pinned":false,"hidden":false,"mode":"ojs"},{"id":265,"value":"tex `P(B|\\text{not } A) = .6`","pinned":false,"hidden":false,"mode":"ojs"},{"id":274,"value":".8*.3/(.8*.3+.2*.6)","pinned":true,"hidden":false,"mode":"ojs"},{"id":26,"value":"html`\n

CSS

\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"html`

Libraries

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":9,"value":"import {slider,button,select,text,radio,checkbox,number} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"math = require('mathjs@6.2.1/dist/math.js')","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"stat = require('simple-statistics@7.3.0/dist/simple-statistics.min.js')\n","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W30","stdlib":"1","titling":"auto"}