38          Binomial Experiments-I
   1         Binomial experiments are those for which the 
              outcome from each trial is one of only two 
              options (“success” or “failure”). The 
              properties of a binomial experiment are:
          1.  The experiment is repeated for a fixed 
              number of trials, where each trial is 
              independent of all the others.
          2.  There are only two possible outcomes for 
              each trial. The outcomes can be classified as 
              a success (S) or a failure (F).
          3.  The probability of success is the same for all 
              trials.
          4.  The random variable X counts the number of 
              successful trials out of n trials.
  38      Binomial Experiments-II
  1     
          Why is the following a binomial 
          experiment?
           We randomly sample 500 fish from 
            the population.
           We record whether each animal is 
            mature or immature.
           The random variable X is the number 
            of mature animals.
               Binomial Experiments-II
  38                       (Notation)
   1
 1. The experiment is repeated for a fixed number of trials
     2. There are only two possible outcomes (S and F)
3. The probability of success P(S)  is the same for each trial
4. The random variable x counts the number of successful trials
        n        The number of times a trial is repeated
 New    p P(S)  The probability of a success in a single trial
 terms qP(F) The probability of a failure in a single trial,q 1- p
        x        The number of successes on n trials (x=0,1,..,n)
    38                    Binomial Probabilities-I
     1             In a binomial experiment, the probability 
                       of exactly x successes in n trials is:
                                           P(x) C pxqnx                   n!       pxqnx
                                                     n   x              (n x)!x!
                   The binomial probability therefore 
                       involves the probability of x successes 
                       and n -x  failures multiplied by the 
                       number of ways choosing x successes out 
                       of n trials.
                   n and p are known as parameters. Much 
                       of statistics involves using data to 
                       estimate the values for unknown 
                       parameters.
  38       Binomial Probabilities-II
  1      Notation:
                     X ~ B(n; p)
          We read this as “The random variable 
          X is distributed binomially with 
          parameters n and p”.
                    Mean: np
                    Variance:2 npq