#set document(title: "5.10 Multinomial Distribution", author: "OpenStax") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 5.10#h(0.6em)Multinomial Distribution #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Prerequisites] Distributions, Basic Probability, Variability, Binomial Distribution #linebreak() #linebreak() ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Learning Objectives] + Define multinomial outcomes + Compute probabilities using the multinomial distribution ] The binomial distribution allows one to compute the probability of obtaining a given number of binary outcomes. For example, it can be used to compute the probability of getting 6 heads out of 10 coin flips. The flip of a coin is a binary outcome because it has only two possible outcomes: heads and tails. The multinomial distribution can be used to compute the probabilities in situations in which there are more than two possible outcomes. For example, suppose that two chess players had played numerous games and it was determined that the probability that Player A would win is 0.40, the probability that Player B would win is 0.35, and the probability that the game would end in a draw is 0.25. The multinomial distribution can be used to answer questions such as: "If these two chess players played 12 games, what is the probability that Player A would win 7 games, Player B would win 2 games, and the remaining 3 games would be drawn?" The following formula gives the probability of obtaining a specific set of outcomes when there are three possible outcomes for each event: #math.equation(block: true, alt: "p equals the fraction n ! over open parenthesis n sub 1 ! close parenthesis open parenthesis n sub 2 ! close parenthesis open parenthesis n sub 3 ! close parenthesis p sub 1 to the power n sub 1 p sub 2 to the power n sub 2 p sub 3 to the power n sub 3")[$p = frac(n !, ( n_(1) ! ) ( n_(2) ! ) ( n_(3) ! )) p_(1)^(n_(1)) p_(2)^(n_(2)) p_(3)^(n_(3))$] where p is the probability, #linebreak() n is the total number of events #linebreak() n#sub[1] is the number of times Outcome 1 occurs, #linebreak() n#sub[2]is the number of times Outcome 2 occurs, #linebreak() n#sub[3] is the number of times Outcome 3 occurs, #linebreak() p#sub[1] is the probability of Outcome 1 #linebreak() p#sub[2] is the probability of Outcome 2, and #linebreak() p#sub[3] is the probability of Outcome 3. #linebreak() For the chess example, n = 12 (12 games are played), #linebreak() n#sub[1] = 7 (number won by Player A), #linebreak() n#sub[2]= 2 (number won by Player B), #linebreak() n#sub[3] = 3 (the number drawn), #linebreak() p#sub[1] = 0.40 (probability Player A wins) #linebreak() p#sub[2] = 0.35(probability Player B wins) #linebreak() p#sub[3] = 0.25(probability of a draw) #math.equation(block: true, alt: "p equals the fraction 12 ! over open parenthesis 7 ! close parenthesis open parenthesis 2 ! close parenthesis open parenthesis 3 ! close parenthesis .40 to the power 7.35 squared .25 cubed equals 0.0248")[$p = frac(12 !, ( 7 ! ) ( 2 ! ) ( 3 ! )) ".40"^(7) ".35"^(2) ".25"^(3) = 0.0248$] #linebreak() The formula for k outcomes is #math.equation(block: true, alt: "p equals the fraction n ! over open parenthesis n sub 1 ! close parenthesis open parenthesis n sub 2 ! close parenthesis and so on open parenthesis n sub k ! close parenthesis p sub 1 to the power n sub 1 p sub 2 to the power n sub 2 and so on p sub k to the power n sub k")[$p = frac(n !, ( n_(1) ! ) ( n_(2) ! ) … ( n_(k) ! )) p_(1)^(n_(1)) p_(2)^(n_(2)) … p_(k)^(n_(k))$] Note that the binomial distribution is a special case of the multinomial when k = 2.