{"id":294,"date":"2017-08-24T07:05:54","date_gmt":"2017-08-24T07:05:54","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=294"},"modified":"2017-08-31T06:14:40","modified_gmt":"2017-08-31T06:14:40","slug":"defining-probability","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/defining-probability\/","title":{"rendered":"Defining Probability"},"content":{"rendered":"
\n

\nWorkshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Defining Probability<\/h1>\n

Objective<\/h4>\n

In this lesson, you will use the concept of probability to discuss common situational probabilities and study some\u00a0helpful counting techniques that are commonly used when calculating probability.<\/p>\n

\n

What is probability?<\/h4>\n

Probability<\/abbr> is the study of how likely something is to happen; more\u00a0specifically, probability helps us decide how likely it is that a\u00a0certain outcome will follow from an event. Think of probability as\u00a0a function that takes one possible outcome from an event and\u00a0returns a number between 0 and 1.<\/p>\n

If the probability of that particular outcome takes the value 0\u00a0then the event is impossible<\/abbr>, meaning it can never happen. If the probability of that particular\u00a0outcome takes the value 1, then the event is guaranteed<\/abbr> to occur, meaning it must happen. Other possible outcomes have\u00a0probabilities that lie somewhere between these extremes.<\/p>\n

\n

Important Tidbit<\/h4>\n

Probability is a numerical value ranging from 0, meaning impossible,\u00a0to 1,\u00a0meaning guaranteed.<\/p>\n

\"Diagram<\/p>\n<\/div>\n

How do we determine probability?<\/h3>\n

As stated earlier, the\u00a0probability function assigns a number between 0 and 1 to any\u00a0particular outcome following from an event. This gives a good\u00a0intuition for determining probability, but how do we actually\u00a0determine the exact number? For any event, we consider the\u00a0possible outcomes, and we determine which of these outcomes are\u00a0<\/span>favorable<\/span><\/em>.\u00a0If an outcome is <\/span>favorable<\/span><\/em>,\u00a0it is not necessarily desirable. It just means that it is the\u00a0outcome for which we are evaluating probability. For example, if\u00a0we are assessing the probability that it will rain, then rain is a\u00a0favorable outcome, regardless of whether we would be happy if it\u00a0rained. <\/span><\/p>\n

First, let’s consider a\u00a0special situation in probability: a situation in which each\u00a0outcome is equally likely, such as rolling a fair die and\u00a0considering the probability of rolling any particular number, or\u00a0drawing a card from a standard deck and considering the\u00a0probability of drawing a particular card. If we let <\/span>E\u00a0<\/span><\/em>stand for <\/span>a\u00a0favorable outcome,\u00a0<\/span><\/em>and <\/span>P<\/span><\/em>(<\/span>E<\/span><\/em>)\u00a0stand for <\/span>the\u00a0probability that we get a favorable outcome<\/span><\/em>,\u00a0then we have the following formula: <\/span><\/p>\n

<\/p>\n

This formula says that the probability of a favorable outcome\u00a0is the ratio of the number of favorable outcomes to total number\u00a0of possible outcomes. The total number of possible outcomes is called the sample\u00a0space<\/abbr>.<\/p>\n

\n

Question<\/h4>\n
\n

Suppose we flip two coins simultaneously. What is the\u00a0probability that <\/span>both\u00a0<\/span><\/em>coins will land\u00a0with their tail sides facing up? <\/span><\/p>\n

    \n
  1. 0<\/li>\n
  2. <\/li>\n
  3. <\/li>\n
  4. <\/li>\n<\/ol>\n<\/div>\n

    Reveal Answer <\/a><\/p>\n

    \n

    The correct choice is B. In this example, E\u00a0<\/span><\/em>= <\/span>both coins land\u00a0with their tail sides facing up<\/span><\/em>.\u00a0To<\/span> use the formula, we must determine the ratio of\u00a0favorable outcomes to total outcomes. If we use H to represent\u00a0heads and T to represent tails, we can see there are four total\u00a0outcomes: HH, HT, TH, TT. Clearly, only one of these four\u00a0outcomes provides the favorable outcome of two tails (TT), and\u00a0thus\u00a0.<\/p>\n<\/div>\n<\/section>\n

    \n

    Be Aware!<\/h4>\n

    Let\u2019s consider another interesting point in computing probabilities. We can reason in the following way: Either a favorable outcome will happen or it won\u2019t. Thus <\/span>E<\/span><\/em> or (not <\/span>E<\/span><\/em>) must occur. Because the probability of a guaranteed event is equal to 1, we get <\/span>P<\/span><\/em>(<\/span>E<\/span><\/em> or (not <\/span>E<\/span><\/em>)) = 1. Furthermore, because <\/span>P<\/span><\/em>(<\/span>E<\/span><\/em> or (not <\/span>E<\/span><\/em>)) = <\/span>P<\/span><\/em>(<\/span>E<\/span><\/em>) + <\/span>P<\/span><\/em>(not <\/span>E<\/span><\/em>), we get <\/span>P<\/span><\/em>(not <\/span>E<\/span><\/em>) = 1 \u2013 <\/span>P<\/span><\/em>(<\/span>E<\/span><\/em>).<\/span><\/p>\n

    Sometimes it is not clear how to count the favorable outcomes and the total outcomes. In this case, it is usually easier to count the outcomes for (not <\/span>E<\/span><\/em>). At times like this, we have to use this equation to compute <\/span>P<\/span><\/em>(<\/span>E<\/span><\/em>) using <\/span>P<\/span><\/em>(not <\/span>E<\/span><\/em>).<\/span><\/p>\n<\/div>\n

    Suppose we let <\/span>E\u00a0<\/span><\/em>represent the outcome “a die lands on a number less than or\u00a0equal to 4.” Then\u00a0.\u00a0The probability that a die lands on either 5 or 6 is\u00a0,\u00a0which is exactly what we would expect from calculating\u00a0\u00a0explicitly.<\/span><\/p>\n

    What does all this mean, really?<\/h3>\n

    As with all mathematics,\u00a0the numerical value of a probability has real-world implications.\u00a0Keeping the formula for probability in mind, we can easily\u00a0interpret the numerical value of a probability measure. Clearly,\u00a0the probability is zero only when there are <\/span>no\u00a0<\/span><\/em>favorable outcomes for an event. Similarly, when each and every\u00a0outcome is favorable, the probability is equal to one. In the\u00a0example above, we found that
    \n.\u00a0Realistically, this means if we flip two coins 100 times, we\u00a0should expect 25 of those flips to return two tails. Or, if we\u00a0flip the two coins four times, we should expect to get two tails\u00a0once. Of course, these are rough estimates. The more trials we\u00a0run, the closer the result will be to 25%. <\/span><\/p>\n

    \n

    Question<\/h4>\n
    \n

    When choosing one card at random from a deck of cards, what is\u00a0the probability of choosing a face card (i.e., a jack, queen, or\u00a0king)? There are 52 cards in a deck and 12 face cards total.<\/p>\n

      \n
    1. <\/li>\n
    2. <\/li>\n
    3. <\/li>\n
    4. <\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct choice is B.\u00a0In this example, <\/span>E\u00a0<\/span><\/em>= “choosing a face card from a single deck.” We are\u00a0only choosing one card, so the number of favorable outcomes is\u00a0equal to the number of face cards in one deck. There are four\u00a0suits, each with three face cards, which means that there are 12\u00a0cards that we can pick to achieve a favorable outcome. There are\u00a052 cards in a deck, so we have 52 total outcomes, therefore\u00a0.
      \n<\/span><\/p>\n<\/div>\n<\/section>\n

      Important Tidbit<\/strong><\/p>\n

      This hierarchy of the possibility of events determines the payoff when gambling. The <\/span>less likely<\/span><\/em> an event is to occur, the <\/span>higher<\/span><\/em> the payoff for the gambler when it does. In poker, the probability of being dealt a hand containing a pair (a hand with any two cards alike) is much greater than the probability of being dealt a royal flush (a hand containing an ace, a king, a queen, a jack, and a ten, all from the same suit). Because a royal flush is so unlikely by comparison, it beats a pair (and everything else for that matter). Therefore, the royal flush holder gets the big payoff. <\/span><\/p>\n

      Isn’t probability just a counting chore?<\/h3>\n

      The formula for calculating probability is quite simple. It\u00a0tells us that probability really just boils down to counting two\u00a0things: the number of favorable outcomes for an event and the\u00a0number of total outcomes. This definition of probability is only\u00a0true when every single outcome is as equally likely as any other.\u00a0If we are to have an accurate count, we need to make sure we do\u00a0not forget any of these outcomes. So far, all the counting we have\u00a0needed to perform has been fairly easy. As one would expect,\u00a0however, some events require much more sophisticated counting\u00a0techniques to determine their probability.<\/p>\n

      The most important of these techniques is the fundamental counting principle<\/abbr>. This principle states that the total number of possible outcomes following from a series of events is determined by multiplying all the ways that each\u00a0individual event can occur.<\/p>\n

      \n

      Question<\/h4>\n
      \n

      Suppose that a license plate consists of two letters followed\u00a0by four digits. The plates use only capital letters, and only the\u00a0numbers 0 through 9 (these letters and numbers can repeat; for\u00a0example, a license plate might feature more than one 3 or more\u00a0than one A). What is the total number of unique license plates\u00a0possible?<\/p>\n

        \n
      1. 452<\/li>\n
      2. 5,000<\/li>\n
      3. 625,000<\/li>\n
      4. 6,760,000<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is D. We have six places that have to be\u00a0filled on the license plate.<\/p>\n

        <\/p>\n

        We can think of each place as an individual event. In each\u00a0blank, we write the number of ways that each place can be filled.\u00a0Assuming we use only capital letters and numbers 0 through 9, we\u00a0have 26 choices for the letter slots and 10 choices for the\u00a0number slots.<\/p>\n

        <\/p>\n

        Using the fundamental counting principle, the total number of\u00a0license plates possible is\u00a0.\u00a0Clearly, the counting principle is a very useful tool.<\/p>\n<\/div>\n<\/section>\n

        When do we use the fundamental counting principle?<\/h3>\n

        Spotting when to use the counting principle can be a tremendous\u00a0help. Counting the ways that a complicated event can occur can be\u00a0tricky. If it isn\u2019t clear how to count the different\u00a0outcomes, try to break the event into a series of events or\u00a0choices. Once you have a series of events from which to choose,\u00a0use the counting principle.<\/p>\n

        Here is another example of the counting principle. Suppose we\u00a0have 7 books to put on a bookshelf and we want to know how many\u00a0different arrangements are possible. We can look at this one event\u00a0of arranging books on a shelf as seven individual events or\u00a0choices.<\/p>\n

        <\/p>\n

        We have 7 books and thus 7 choices for the first slot.<\/p>\n

        <\/p>\n

        Because only six books are left after the first slot is filled,\u00a0we have six choices for the second slot.<\/p>\n

        <\/p>\n

        Continuing this way and multiplying, we find that there are<\/p>\n

        \u00a0ways of arranging seven books on the shelf. Recall\u00a0that the <\/span>!\u00a0<\/span><\/em><\/strong>operation is called the <\/span>factorial<\/span><\/abbr>.<\/span><\/p>\n

        \n

        Important Tidbit<\/h4>\n

        and is read \u201c<\/span>n<\/span><\/em> factorial.<\/span>\u201d<\/p>\n<\/div>\n

        \n

        Be Aware!<\/h4>\n

        When computing factorials it is important\u00a0to remember\u00a0the following facts:<\/p>\n

        The factorial is defined <\/span>only<\/span><\/em> for nonnegative numbers (0, 1, 2, 3, . . . ). <\/span><\/p>\n

        So you should never have to compute \u20137! or \u20133!.\u00a0These computations do not make sense. If they arise, check your work again to see if you made a\u00a0mistake.<\/p>\n

        For notational purposes, 0! = 1.<\/p>\n<\/div>\n<\/section>\n

        <\/p>\n


        \n        Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

        Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

        Workshop Index\u00a0Next Lesson \u27a1 Defining Probability Objective In this lesson, you will use the concept of probability to discuss common situational probabilities and study some\u00a0helpful counting techniques that are commonly used when calculating probability. What is probability? Probability is the study of how likely something is to happen; more\u00a0specifically, probability helps us decide how likely […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-294","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/294","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=294"}],"version-history":[{"count":8,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/294\/revisions"}],"predecessor-version":[{"id":625,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/294\/revisions\/625"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}