Transcript Probability

Probability
Section 7.1
What is probability?
Probability discusses the likelihood or chance of
something happening. For instance,
-- the probability of it raining tomorrow
-- the probability of rolling a 4 on a six-sided die
-- the probability of being struck by lightning
-- the probability of pulling an Ace out of a deck of cards
-- the probability of meeting your spouse on a blind date
We will try to answer some of these questions in these
chapters. Let’s start with a relatively easy example.
Consider rolling a six-sided die. What is the probability of
getting a 4 on the die? Notice the notation we will use.
1
P ( 4) 
6
We can think of this as the
ratio of successes to total
possibilities. There is 1
success (the 4) on a die
and 6 possible sides.
First, this does assume all sides are equally likely to come
up when I roll the die. This is usually the case when we
talk about dice.
This formula for probability will come in handy. Keep it in
mind.
We should cover some terminology that we will use
throughout our work here. Examples are given after each
term.
Terminology
experiment: the act you do (roll a die)
outcome: any one possible result (1, 2, 3, 4, 5, or 6)
sample space: the set of all possible outcomes
( {1, 2, 3, 4, 5, 6} )
event: a particular set of outcomes ( {2, 4, 6} or evens)
trial: one instance of the experiment (If you roll a die 50
times, you’ve done 50 trials.)
success: specific event you’re interested in (If we want the
probability that the die will be even, we have three
successes; they are 2, 4, and 6.)
Theoretical probability: the probability of an event based
on the context of the problem
Experimental probability: the probability of an event
based on doing the experiment many times
Consider rolling a die. The theoretical probability of
getting an even number is
P(even)  3
6 . This is
gotten by thinking about the die and figuring that there
are 3 successes and 6 total possibilities.
If we were to roll a die 100 times and actually roll an even
number 47 of the times, our experimental probability
would be
P(even)  47
.
100
Worksheets
“Probability applet activity” investigates the relationship
between theoretical and experimental probability via an
online activity. It simulates the tossing of a coin many
times. You are looking at the proportion of heads.
“The difference between OR and AND” covers some
fundamentals needed to understand probability.
Consider the experiment
Roll die, toss coin.
What’s P(4 and H)?
Sample space:
1T
2T
3T
4T
5T
6T
1H
2H
3H
4H
5H
6H
You need to
write the
outcomes
down in an
orderly fashion.
Now calculate
There are 12 possibilities
and 1 success…
So P(4 and H) = 1/12. Let’s look at it a different way.
Notice P(4 on die) = 1/6
and P(H on coin) = ½.
And
16 12  112
P(4) P( H )  P(4andH)
This is an example of a commonly used rule. We’ll state it
in general but we need a definition first.
Independent events
Two events are independent if the occurrence of one does
not affect the occurrence of the other. The events “4” and
“H” are independent because the occurrence of the 4 on
the die does not affect the probability that I will get an H
also.
Rule: If events E and F are independent, then the
probability that they both occur is equal to the probability
that E occurs times the probability that F occurs or
P( EandF)  P( E ) P( F ).
Mutually exclusive events
Two events are mutually exclusive if they cannot happen
at the same time.
expl: experiment: roll die
A: roll even number
B: roll a 3
Events A and B are mutually exclusive because you cannot
roll an even number and a 3 at the same time.
Worksheet
“Probability Worksheet 2” covers the four types of problems
we will encounter. They are finding the probability that
1.) either of two mutually exclusive events occur,
2.) either of two non-mutually exclusive events occur,
3.) two independent events both occur, and
4.) two non-independent events both occur.
Its last page contains some practice problems.
“Solutions to Probability worksheet 2” is available. It
explains the practice problems at the end of the
worksheet.
Probability Worksheet 2
There are four rules that are discussed on this worksheet.
They are summarized here.
1.) If A and B are mutually exclusive,
then P(A or B) = P(A) + P(B).
2.) If A and B are not mutually exclusive,
then P(A or B) = P(A) + P(B) – P(A and B).
3.) If A and B are independent,
then P(A and B) = P(A)  P(B).
4.) If A and B are not independent,
then P(A and B) = P(A)  P(B given A).
Some questions
1.) What is the lowest a probability can be? Can it be 0 or
negative?
2.) What is the highest a probability can be? Can it be 2?
3.) What is the sum of the probabilities of all possible
mutually exclusive events?
Use the experiment of rolling a die 100
times. What is the lowest P(4) can be?
What is the highest? What is
P(1) + P(2) + P(3) + P(4) + P(5) +
P(6)?
Some answers
Since experimental probability is
, and
since the number of successes cannot be negative but it
could be zero, the lowest a probability can be is 0.
Since the number of successes cannot exceed the number
of trials, the highest a probability can be is 1.
We see that P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1.
Since these are the only six things that can occur and
one must occur, then the sum of the probabilities of the
possible mutually exclusive events must be 1.
Terminology
An event that has probability 1 must always happen. It is
called a sure or certain event.
experiment: Toss coin
event: heads or tails
When you toss a coin, you must get either a heads or a tail.
An event that has probability 0 will never happen. It is
called an impossible event.
experiment: Roll die
event: roll 7
When you roll a six-sided die, you cannot get a 7.
Complement of an event
If we let A represent an event, then A is used to represent
its complement.
The complement of an event is made up of the outcomes
from the sample space that are not in the original event.
Consider pulling a single card out of a deck of poker cards.
Let A represent the event “red Queen”. Here, A can be
interpreted as “the Queen of Hearts or the Queen of
Diamonds”.
The complement of A would be all of the other 50 cards.
Since either an event occurs or it does not, A and A
must be mutually exclusive; they cannot happen at the
same time. Since either A or A must happen, we know
P( A)  P( A)  1 .
A more complicated example
Consider the experiment of rolling two distinguishable sixsided dice. We are interested in finding the probability
that the sum of the two dice rolled is 6. Let’s look at the
sample space (36 events) of this experiment.
In finding the probability of rolling two dice whose sum is 6,
we need to think about
Since there are 5 successes out of 36 equally likely
5
possibilities, the probability of rolling a sum of 6 is 36 .
Worksheets
“Probability experiment” is a group activity where you will
think up a probability question and find its experimental
probability.
“Mutually exclusive and independent events” investigates
the relationship between the qualities mutually exclusive
and independent.
Other optional worksheets are available online.
Homework
Remember, most problems can be
handled using the basic definition,
number of successes divided by
number of possibilities. Use the
four rules discussed to do more
complicated problems.
7.1: 1, 2, 3, 5, 7, 8, 9, 11, 12, 22, 24, 26, 27, 28, 29, 31, 34,
37b