{"id":2122,"date":"2015-11-12T18:30:41","date_gmt":"2015-11-12T18:30:41","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2122"},"modified":"2015-11-12T18:30:41","modified_gmt":"2015-11-12T18:30:41","slug":"computing-the-probability-of-mutually-exclusive-events","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/computing-the-probability-of-mutually-exclusive-events\/","title":{"raw":"Computing the Probability of Mutually Exclusive Events","rendered":"Computing the Probability of Mutually Exclusive Events"},"content":{"raw":"

Suppose the spinner in Figure 2\u00a0is spun again, but this time we are interested in the probability of spinning an orange or a [latex]d[\/latex]. There are no sectors that are both orange and contain a [latex]d[\/latex], so these two events have no outcomes in common. Events are said to be mutually exclusive events<\/span> when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is\n<\/p>

[latex]P\\left(E\\cup F\\right)=P\\left(E\\right)+P\\left(F\\right)[\/latex]<\/div>\nNotice that with mutually exclusive events, the intersection of [latex]E[\/latex] and [latex]F[\/latex] is the empty set. The probability of spinning an orange is [latex]\\frac{3}{6}=\\frac{1}{2}[\/latex] and the probability of spinning a [latex]d[\/latex] is [latex]\\frac{1}{6}[\/latex]. We can find the probability of spinning an orange or a [latex]d[\/latex] simply by adding the two probabilities.\n
[latex]\\begin{array}{l}P\\left(E{\\cup }^{\\text{ }}F\\right)=P\\left(E\\right)+P\\left(F\\right)\\hfill \\\\ \\text{ }=\\frac{1}{2}+\\frac{1}{6}\\hfill \\\\ \\text{ }=\\frac{2}{3}\\hfill \\end{array}[\/latex]<\/div>\nThe probability of spinning an orange or a [latex]d[\/latex] is [latex]\\frac{2}{3}[\/latex].\n
\n

A General Note: Probability of the Union of Mutually Exclusive Events<\/h3>\nThe probability of the union of two mutually exclusive<\/em> events [latex]E\\text{and}F[\/latex] is given by\n
[latex]P\\left(E\\cup F\\right)=P\\left(E\\right)+P\\left(F\\right)[\/latex]<\/div>\n<\/div>\n
\n

How To: Given a set of events, compute the probability of the union of mutually exclusive events.<\/h3>\n
  1. Determine the total number of outcomes for the first event.<\/li>\n\t
  2. Find the probability of the first event.<\/li>\n\t
  3. Determine the total number of outcomes for the second event.<\/li>\n\t
  4. Find the probability of the second event.<\/li>\n\t
  5. Add the probabilities.<\/li>\n<\/ol><\/div>\n
    \n

    Example 4: Computing the Probability of the Union of Mutually Exclusive Events<\/h3>\nA card is drawn from a standard deck. Find the probability of drawing a heart or a spade.\n\n<\/div>\n
    \n

    Solution<\/h3>\nThe events \"drawing a heart\" and \"drawing a spade\" are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is [latex]\\frac{1}{4}[\/latex], and the probability of drawing a spade is also [latex]\\frac{1}{4}[\/latex], so the probability of drawing a heart or a spade is\n
    [latex]\\frac{1}{4}+\\frac{1}{4}=\\frac{1}{2}[\/latex]<\/div>\n<\/div>\n
    \n

    Try It 4<\/h3>\nA card is drawn from a standard deck. Find the probability of drawing an ace or a king.\n\nSolution<\/a>\n\n<\/div>","rendered":"

    Suppose the spinner in Figure 2\u00a0is spun again, but this time we are interested in the probability of spinning an orange or a [latex]d[\/latex]. There are no sectors that are both orange and contain a [latex]d[\/latex], so these two events have no outcomes in common. Events are said to be mutually exclusive events<\/span> when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is\n<\/p>\n

    [latex]P\\left(E\\cup F\\right)=P\\left(E\\right)+P\\left(F\\right)[\/latex]<\/div>\n

    Notice that with mutually exclusive events, the intersection of [latex]E[\/latex] and [latex]F[\/latex] is the empty set. The probability of spinning an orange is [latex]\\frac{3}{6}=\\frac{1}{2}[\/latex] and the probability of spinning a [latex]d[\/latex] is [latex]\\frac{1}{6}[\/latex]. We can find the probability of spinning an orange or a [latex]d[\/latex] simply by adding the two probabilities.<\/p>\n

    [latex]\\begin{array}{l}P\\left(E{\\cup }^{\\text{ }}F\\right)=P\\left(E\\right)+P\\left(F\\right)\\hfill \\\\ \\text{ }=\\frac{1}{2}+\\frac{1}{6}\\hfill \\\\ \\text{ }=\\frac{2}{3}\\hfill \\end{array}[\/latex]<\/div>\n

    The probability of spinning an orange or a [latex]d[\/latex] is [latex]\\frac{2}{3}[\/latex].<\/p>\n

    \n

    A General Note: Probability of the Union of Mutually Exclusive Events<\/h3>\n

    The probability of the union of two mutually exclusive<\/em> events [latex]E\\text{and}F[\/latex] is given by<\/p>\n

    [latex]P\\left(E\\cup F\\right)=P\\left(E\\right)+P\\left(F\\right)[\/latex]<\/div>\n<\/div>\n
    \n

    How To: Given a set of events, compute the probability of the union of mutually exclusive events.<\/h3>\n
      \n
    1. Determine the total number of outcomes for the first event.<\/li>\n
    2. Find the probability of the first event.<\/li>\n
    3. Determine the total number of outcomes for the second event.<\/li>\n
    4. Find the probability of the second event.<\/li>\n
    5. Add the probabilities.<\/li>\n<\/ol>\n<\/div>\n
      \n

      Example 4: Computing the Probability of the Union of Mutually Exclusive Events<\/h3>\n

      A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      The events “drawing a heart” and “drawing a spade” are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is [latex]\\frac{1}{4}[\/latex], and the probability of drawing a spade is also [latex]\\frac{1}{4}[\/latex], so the probability of drawing a heart or a spade is<\/p>\n

      [latex]\\frac{1}{4}+\\frac{1}{4}=\\frac{1}{2}[\/latex]<\/div>\n<\/div>\n
      \n

      Try It 4<\/h3>\n

      A card is drawn from a standard deck. Find the probability of drawing an ace or a king.<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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