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MIDDLE GROUND - Binomial Formula Explained

I.   Brief Summary of A Binomial Distribution
0.   Basic Probability and Counting Formulas
      Vocabulary, Facts, Count the Ways to Make An Ordered List Or A Group
      The average is the sum of the products of the event and the probability of the event.
II.   Binomial Distribution Explained More Slowly
III. Binomial Formula Explained
      Combinations Compute The Number of Each Outcome in A Binomial Distribution
      What's the Probability of Obtaining Exactly 3 Heads If A Fair Coin Is Tossed 4 Times?
IV. Sum of the Probabilities and the Binomial Mean
      The Sum of The Probabilities Is One.
      The Expected Value Is The Mean.
      The Mean, Expected Value, Is (n)(p).
      Why the mean, expected value, is (n)(p)
V.   Examples
VI. Use the Normal to Compute the Binomial on a Calculator
Use the Normal to Compute the Binomial.
      Using the normal to compute the binomial is an important mathematical idea but now an old computational technique. For the most part, calculators functions have replaced statistical tables as sources for values of probability distributions.
      It is valuable to know that for large values of n, the discrete binomial distribution approaches the continuous normal distribution.
      When n(p) >5 and n(q) >5, the normal distribution may be used to approximate the binomial. This technique is useful when:
  • binomial tables are not available for the given n,
  • the numbers for n is large, or
  • computation is required for more than 1 value of x, as in p(x>2), since repeated use of the formula or even the tables might be a pain.
      The normal distribution does not match the binomial distribution, but, for larger values of n the shape matches closer and closer.
      In each distribution, the probability is the area under the curve. For the binomial distribution (which is discrete), each value of x has a thickness on the x axis. For the normal distribution (which is continuous), each value of x has one one point on the x-axis where a vertical line crosses the axis.
      Since a line does not have an area (the vertical line x=2 to compute p(x=2), for example), one must use an interval (the interval 1.5 < x < 2.5 to compute p(1.5 < x < 2.5), for example) to compute the probability.
      Because of the above, to use this technique, one must rewrite the binomial probability expression to match an expression which can be evaluated by a normal probability expression.
      This technqiue requires taking the statement of equality (one which inlcudes =) and rewriting it as a statement of inequality (one which includes >, <, >, or <, or multiple statements of inequality. The figures below depict this technique.
Compute Using A Calculator
      The Distribution menu, [DISTR], lists probability distribution functions. It is found above the Varibles menu, [2nd] [VARS].
      Funtion A, listed as [A:binomcdf(], is the cumulative BINOMIAL distribution. It is similar to the NORMAL cumulative distribution funtion, [2:normalcdf(]. It adds "from the left."
      Each function requires parameters, stated in specific order and manner, for the function to work.
      If p is the probability of success and x is the number of successes on n trials, the cumulative binomial distribution is computed using binomcdf(n,p,x) where , is the comma key, [,].
      To compute p(x<1) where n is 4 and p is .4, use
      This value is .4752 and is shown below on a calculator home screen and in a written-upon binomial distribution table.

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