FINDING BINOMIAL PROBABILITIES

~In a two-case outcome category (yes, no), (heads, tails), (for, against), & so on, one of the outcomes is considered a success (this is an arbitrary choosing) & the other a failure. 

~If a set number of trials are done (value of n) & the probabilities remain the same for each trial, we can find probabilities of an exact number of successes or a range of successes. 

~These are calculated by using the binomial formula as follows:

   nCr p^rq^(n-r),   Remember: q = 1- p

Where:

 n=the # of trials
 C is the combination symbol
 r = the # of successes
 p = the probability of a success
 q = the probability of a failure .

~Combinations can be found on the TI-83 at the MATH button, then  PRB on the top, then to menu 3. However, before you go to this menu, you must enter n first.

~Using the TI-83, this calculation is made simple. Just go to: 
  2nd, VARS, down to Menu O, enter (n, p, r), enter.
  (Use this menu only for AN EXACT # of successes)

~For problems which involve "AT LEAST", "UP TO", "AT MOST", "LESS THAN", & "GREATER THAN", a range of successes need to be found. Menu O should not be used. (Unless you want to calculate the different number of successes independently then add them. (Highly impractical in most cases)    

~Here we use MENU A located right after menu O. This menu gives the CUMULATIVE probabilities (i.e., from 0 up to a given number, r). So, it can be used directly for problems involving "AT MOST r out of n SUCCESSES" or "NO MORE THAN r SUCCESSES", or "r OR FEWER", since all of these are cumulative. Just enter (n,p,r) in that order, then press enter.

~For problems dealing with "AT LEAST r" , "r OR MORE", or "MORE THAN r", it becomes a little tricky. Menu A CANNOT be  used directly since these are not cumulative. To transform them to cumulative, one uses the BACKDOOR APPROACH to probability. That is:

    P(A) = 1 - P(THE  COMPLEMENT OF A)

So, one must know the complement of  "AT LEAST r".
That would be  "AT MOST" r-1 (which is cumulative & menu A can be use for that part). Also, the complement of "MORE THAN r" would be "AT MOST" r successes, which becomes cumulative).

~However, many errors are mand by students because they forget to subtract this probability from one.  This error can be avoided by entering  1 - P(complement of A) directly into the TI-83, so that the subtraction does not have to be made separately.

~For the problem involving "at least r successes in n trials", enter 
1 - then go to 2nd VARS, go to menu A, enter (n, p, r-1), enter.  A similar procedure is used for "more than r successes in n trials", just enter (n, p, r) for the complement.

~See my articles on "Pascal's triangle" & "Bernoulli's backdoor" under
 "topics of interest" on this web site. Also, see the link, "Popular menus on the TI-83 in Statistics".

~NOTE:  Menus O & A on the TI-83 are located in different places on the TI-84 (menus A and B). Also, some newer TI-83 calculators have them located in similar places. Look for binompdf & binomcdf on the TI-84.