Page 51 - Math SL HB Sem 3
P. 51

Cumulative  binomial probability  :

                 To find the cumulative  binomial  probability,  you need to first determine which probabilities are to
                 be summed  up. For example,
                                                  P(x 3 2)  -  P(x  =  0) + P(x  t) + P(x  =  2)
                                                                              =
                                                  P(x<2)- P(x= 0)+P(x=1)
                    Note that the type of
                    the inequality  sign          P(x>2)= 1-P(x<1)
                    determines which
                    probahil  ities should b e
                                                           : 1-
                    added-                                        [P(x =  0) + P(x  =  1)]
                                                  P(x>2)- r-P(x<2)


                                                           =  1  - P(x =  0) + P(x = r) + P(x  -  2)l

                          Watch  Out:

                   P(X  =  0) may not be zero. Don't miss out this term!


                 GDC


                        Calculator-T-:  'DIST'-  'binomcdf  (n, p  ,0' ; Calculator-G-  : call out 'Binomial  c.d.'
                        E.g.  To find
                        P(X < 2) t  enter the upper  limit, r (or 'x'  for Calculalorc)  =  2,,
                                   the number  of trails  ,  n, and the probability  of success,  p, accordingly.
                        P(X > 2)  ,gnter  the upper  limit  , r (or'x'for Calculator-C)  =  1,
                                   the number  of trails  , n, and the probability  of success,  p, accordingly.
                                                                      -
                                   Then subtract  the answer  from  1. (i.e. 1   binomcdf (n,p  ,r -  l))

                  \. Example:


                  Amy tossed  a fair coin eight times. Find the probability  that she obtains
                   a) exactly 4 heads
                   b) at least 3 heads

                  \  sotution:
                                                                                      ldentify  the values  of n and
                                                                                      p first:
                  a)                                              1
                                                          x-B (8,                     There are B trials, so
                                                                  2                   n =8.
                                                                                      It is a fair coiL so
                                                  P(x=a)=                             p=P(head)=0.5
                   Always write down  what                    0()'()'-'
                   probability  that you are
                   finding in the form of                         7
                   P(x -  x)                               =70    t6 x+)
                                                              35
                                                                  (= 0.273)
                                                             t28
                  b)                          =1-P(x<2)
                                      flx>:)
                                              =  -          -     =  L)  P(x  =  2)
                                                                       -
                                                t  P(x  =  0)  P(x
                                              = ,- ()             (?)()'()'-  ()'()"
                      Spot the word'at leas(, and      H'(:)'-                  0
                      form the prcbability           117
                      P(X > x).                                                       It is faster to do
                                                    256 32  64
                                                                                       -
                                                                                      1  P(X < 2)  than to sum
                                                219                                    P(x=3)+P(X=4)
                                                256                                   +P(x=s)+P(x=6)
                                                                                      +P(x ='t) + P(x  =  B)
                                              =  0.855Gon""t  toz
                                                              "ie.tig:)
                                                              tl
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