{"id":1603,"date":"2024-05-14T22:51:23","date_gmt":"2024-05-15T02:51:23","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&p=1603"},"modified":"2024-06-12T14:47:29","modified_gmt":"2024-06-12T18:47:29","slug":"at-most-and-at-least","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/at-most-and-at-least\/","title":{"raw":"Calculating At Least, At Most and More Than 'X' Events","rendered":"Calculating At Least, At Most and More Than ‘X’ Events"},"content":{"raw":"
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nCalculate the probability of at least, at most or more than a certain number of events, [latex]x[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n

Calculating the Probability of At Most 'X<\/em>' Events Occurring<\/h2>\r\n
    \r\n \t
  1. Use the formula<\/a> to calculate each probability up to and including x<\/span>: [latex]P(X\\le x) = P(X=0)+P(X=1)+...+P(X=x)[\/latex]<\/li>\r\n \t
  2. Use cumulative = [latex]TRUE[\/latex] in Excel<\/span>: [latex]P(X\\le x)[\/latex] = POISSON.DIST<\/span>([latex]x[\/latex], [latex]\\lambda[\/latex], TRUE)<\/li>\r\n<\/ol>\r\nWhich is better?<\/em> It is often much quicker to use Excel's POISSON.DIST<\/a><\/span>\u00a0than the formula<\/a>.\r\n

    Calculating the Probability of More Than 'X<\/em>' Events Occurring<\/h2>\r\n
      \r\n \t
    1. Use the formula and a complement to calculate<\/span>: [latex]P(X\\gt x) = 1-[P(0)+P(1)+...+P(x)][\/latex]<\/li>\r\n \t
    2. Use a complement and [latex]TRUE[\/latex] in Excel<\/span>: [latex]P(X\\gt x) =1\u2212[\/latex]POISSON.DIST<\/span>([latex]x[\/latex], [latex]\\lambda[\/latex], TRUE)<\/li>\r\n<\/ol>\r\nWhy use a complement<\/em>? Since there is no upper limit to the number of events that can occur, [latex]P(X\\gt x) = P(x+1)+P(x+2)+...+P(\\infty)[\/latex]. We cannot perform this calculation without using limit theory. Instead, we use a complement of up to the probability of [latex]x[\/latex] events occurring.\r\n

      Calculating the Probability of At Least 'X<\/em>' Events Occurring<\/h2>\r\n
        \r\n \t
      1. Use the formula and a complement<\/span>: [latex]P(X\\ge x) = 1-[P(0)+P(1)+...+P(x-1)][\/latex]<\/li>\r\n \t
      2. Use a complement and [latex]TRUE[\/latex] in Excel<\/span>: [latex]P(X\\ge x)[\/latex] =\u00a01\u2212 POISSON.DIST<\/span>([latex]x-1[\/latex], [latex]\\lambda[\/latex], TRUE)<\/li>\r\n<\/ol>\r\nWhy do we use <\/em>[latex]x-1[\/latex]\u00a0instead of<\/em> [latex]x[\/latex]? Since [latex]x[\/latex] is included in the interval, it needs to be excluded when taking the complement. Ie: [latex]P(X\\ge x) = 1-P(X \\lt x) = 1-P(X \\le x-1)[\/latex]\r\n

        'At MOST' Example (VIDEO)<\/h1>\r\n