Download Counting and Probability

CSRU 1100
Counting and Probability
Counting is Based on Straightforward Rules
• Are countable items combined using the
terms such as AND or OR?
• Are countable items orderable and if so does
the order matter in the particular case?
• Do items get reused when you count, or does
the use of one item decrease the number of
possibilities of the next item?
Counting Can be Summarized as Follows
• Rule #1: If you can count them on your own, then
count them.
• Rule #2: If terms combine with “OR” then you add
the numbers.
• Rule #3: If terms combine with “AND” then you
multiply the numbers.
• Rule #4: If the order you select the numbers does not
matter (but there is a scenario where they could
matter) then divide your answer by n! where n is the
numbers of items you are selecting.
Example
Picking Cards
• If you have 52 cards in a deck. How many different
ways could someone be dealt a 5 card hand that
contains 4 Aces.
• You are selecting 5 cards and the order does not
matter.
– You are going to be dealt 4 Aces and you are going to be
dealt a 5th card.
– 4 * 3 * 2 * 1 (but order does not matter so divide by 4!)
– 48 choices for the 5th card.
– 1 * 48 = 48
Example
Picking License Plates
• Some states have license plates formed with two
letters (which must be different) followed by 4 letters
or numbers (which can be the same. How many
license plates possibilities are there.
• Pick two different letters AND pick 4
letters/numbers.
• Order matters in both cases.
• 26*25 * 36*36*36*36 = 1091750400
Probability
• Knowing how to count also gives you the ability to
compute the probability of some event.
• General rules about probability
– All probabilities are numbers between 0 and 1
– A probability of 1 means something is absolutely going to
happen
– A probability of 0 means something is NOT going to
happen
Probability is just counting
(Twice)
• Each probability is two counting problems.
– Determine how many possibilities you are
interested in having occur (this is called the set of
outcomes).
– Determine how many total possibilities of some
general event (this is called the sample space)
– Divide the first number by the second – this is
your probability
Example
Horse Racing
• 21 horses are in the Kentucky Derby. What is
the probability of you picking the winner?
– There is only 1 outcome that interests you (the
horse you picked winning)
– There are 21 total possible outcomes (each horse
could potentially win)
– Probability is 1/21
Example
Horse Racing 2
• What is the probability that you can pick the
top three finishers in order?
– Well again, there is only 1 order that interests you.
– There are 21*20*19 different possibilities for the
top three to finish (since order matters).
– 1/7980
Example
Electing Class Officers
• If I am going to select 3 people at random from a
class of 20 to be president, vice-president and
secretary. What is the probability that you are one of
the three students.
– How many groups of 3 are you part of?
•
•
•
•
•
There are 19*18*1 ways you could be secretary
There are 19*1*18 ways you could be VP
There are 1*19*18 ways you could be president
You could be President OR VP OR Secretary.
1026 different groupings you could be part of
Class Officers (cont)
• How many total groups of 3 are there (order
matters)
– 20*19*18 = 6840
• Probability that you are in one of the groups is
• 1026/6840 = .15
Example
Card Example Revisited
• What is the probability of being dealt a 5-card hand
that contains 4 aces.
• We know from earlier that there are 48 different
hands with 4 aces.
• How many different 5 card hands are there (order
does not matter)
52 * 51 * 50 * 49 * 48 / 5! = 2598960
• So your probability of getting 4 aces is 48/2598960
Trick about order mattering
• When doing probabilities the order mattering
question ultimately goes away.
• As long as you are consistent between what you do
with the outcome space and the sample space it
won’t matter if you make the wrong decision about
order mattering.
• In other words as long as you do the same thing for
both the outcome space and the sample space then
the ordering info cancels itself out.
Other Ideas
• When you look at each possible outcome of
an event and determine its probability you will
discover that all of the probabilities always
add up to 1.
• What are the outcomes of flipping a coin
– Heads – probability ½
– Tails – probability ½
– They add up to 1