Addition Principle: Key thing
no two of the procedures may be performed simultaneously.
procedure has n1 + n2+… possible outcomes
Find the probability of obtaining exactly one 6 when six fair dice are rolled
number of possible outcomes = 6^6
Let E = ‘exactly one 6 is obtained’. Then E must be
achieved by one of the following procedures:
Using MP, the number of outcomes consistent with 1
(scoring a 6 on the first die but no 6 on any other die) is:
n1 = 1 x 5 x 5 x 5 x 5 x 5 = 5^5 = 3125
but there are 6 dice, so we have 6(3125)
P(E) = 18750/46656
suppose that r objects are to be chosen out of n objects and the order is important:
n!/(n-r)! = n(n-1)….(n-r+1)
8 athletes run a 100m sprint. The outcome of the race will
be a list of the first three athletes, in the order that they
finish. How many different possible outcomes are there?
8 x 7 x 6 = 336
or
8!/5!
total number of DISTINCT permutations of n objects, of n1 which are alike of one kind, another n2 are alike of a second kind, …. n1
n! / n1!n2!,n3!
4 Americans, 3 Britons and 1 Canadian run a 100m sprint.
The outcome of the race will be a list of the countries to
which the athletes belong, in the order that they finish.
How many different possible outcomes are there
8! / 4! 3! = 280
suppose now that r out of n objects are to be chosen, but ORDER IS NOT IMPORTANT. This is called a combination
n!/r!(n-r)! = N choose R
A clinical trial is to be conducted using a group of 20
patient volunteers. 10 of them will be selected at random
to receive a new treatment, and the remaining 10 will
receive the standard treatment. If there are 6 women and
14 men among the volunteers, find the probability that
exactly 3 women and 7 men will receive the new
treatment.
sample space = (20 10)
women = (6 3)
men = (14 7)
MP = (6 3) (14 7)
prob = (6 3)(14 7) / (20 10)
