Transcript The Fundamental Counting Principle

On New Year's Eve, the probability of a person
having a car accident is 0.09. The probability of
a person driving while intoxicated is 0.32 and
probability of a person having a car accident
while intoxicated is 0.15. What is the
probability of a person driving while intoxicated
or having a car accident?
Permutations
and
Factorials!!!!!!!!!!
Determing the number of ways n
objects can be arranged.
Burger or a Taco = 2 selections
Soup or Salad = 2 selections
Ice Cream, Cake or Pie = 3 selection
2 × 2 × 3 = 12
12 possible combinations
An ordered arrangement of objects.
The number of different permutations of
n distinct objects is n!
n factorial
𝑛! = 𝑛 ∙ 𝑛 − 1 ∙ 𝑛 − 2 ∙ (𝑛 − 3) … . .
0! = 1
1! = 1
2! = 2 ∙ 1 = 2
3! = 3 ∙ 2 ∙ 1 = 6
4! = 4 ∙ 3 ∙ 2 ∙ 1 = 24
Finding the number of permutations of n
Objects
The number of permutations is 9!
9x8x7x6x5x4x3x2x1=
362,880
𝑛!
𝑛𝑃𝑟 =
𝑛−𝑟 !
Use this formula when the order matters but
digits can not repeat.
𝑛𝑟
Use this formula when the order matters but
digits can repeat.
Think about finding 6 numbers on a license
plate choosing 1-9
How many different combinations can you make if you can repeat
numbers?
Think about finding 6 numbers on a license
plate choosing 1-9
9x9x9x9x9x9
96
531,441 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠
𝑛𝑟
Think combinations for your school locker
there are 50 umbers to choose from and only 3
numbers are used.
How many different combinations can you make if you
can repeat numbers?
Think combinations for your school locker
there are 50 umbers to choose from and only 3
numbers are used.
50x50x50
503
𝑛𝑟
125,000 combinations
Think combinations for your school locker
there are 50 umbers to choose from and only 3
numbers are used.
How many different combinations can you make if you
can NOT repeat numbers?
Think combinations for your school locker
there are 50 umbers to choose from and only 3
numbers are used.
𝑛!
𝑛𝑃𝑟 =
𝑛−𝑟 !
50!
50𝑃3 =
50 − 3 !
50!
50𝑃3 =
47!
50𝑥49𝑥48𝑥47𝑥46𝑥45
50𝑃3 =
47𝑥46𝑥45𝑥44
50 𝑥 49 𝑥 48
117, 600 Combinations
Suppose we toss a gold dollar coin 8 times.
What is the probability that the sequence of
8 tosses yields 3 heads (H) and 5 tails (T)?
Solution. Two such sequences, for example, might
look like this:
HHHTTTTT
or this
HTHTHTTT
Assuming the coin is fair, and thus that the
outcomes of tossing either a head or tail are equally
likely, we can use the classical approach to
assigning the probability. The Multiplication
Principle tells us that there are:
2×2×2×2×2×2×2×2
256 possible outcomes in the sample space of 8
tosses.
Now, when counting the number of sequences of 3
heads and 5 tosses, we need to recognize that we
are dealing with arrangements or permutations
since order matters, but in this case not all of the
objects are distinct.
We can think of r = 3 positions for the heads (H)
out of the n = 8 possible tosses. That would, of
course, leave then n− r = 8 − 3 = 5 positions for
the tails (T). Using the formula for a combination
of n objects taken r at a time, there are therefore:
8
8!
=
= 56
3
3! 5!
Suppose Dr. Dre conducts research which
involves infecting 20 people with the swine flu
virus. He is interested in studying how many
actually end up ill (I) and how many remain
healthy (H).
How many arrangements are there of the 20
people that involve 0 people ill?