Download Probability - WordPress.com

Probability
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
• 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
•
•
P(even)  3
getting an even number is
6
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
Counting Sample points
As a further illustration, we may be interested in the event B that the number of
defectives is greater than 1 in previous Example. This will occur if the outcome is an
element of the subset
B = {DDN,DND,NDD,DDD} of the sample space S.
To each event we assign a collection of sample points, which constitute a subset
of the sample space. That subset represents all of the elements for which the event
is true.
Venn Diagram
Consider the experiment
You need to
write the
outcomes
down in an
orderly fashion.
• Roll die, toss coin.
• What’s P(4 and H)?
• Sample space: 1T
•
•
•
•
•
1H
2T
3T
4T
5T
6T
2H
3H
4H
5H
6H
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 iff 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.
Exhaustive Events
• Events are said to be collectively exhaustive events
when the union of mutually exclusive events is the
entire sample space S.
Worksheet
• “Probability Worksheet ” 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 ” is available. It explains
the practice problems at the end of the worksheet.
•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
•
possibilities, the probability of rolling a sum of 6 is
5
36
Rules for finding Probability
Suppose the manufacturer specifications of the length of a certain type of
computer cable are 2000 ± 10 millimeters. In this industry, it is known that
small cable is just as likely to be defective (not meeting specifications) as
large cable. That is, the probability of randomly producing a cable with
length exceeding 2010 millimeters is equal to the probability of producing a
cable with length smaller than 1990 millimeters. The probability that the
production procedure meets specifications is known to be 0.99.
(a) What is the probability that a cable selected randomly is too large?
(b) What is the probability that a randomly selected cable is larger than 1990
millimeters?
Solution: Let M be the event that a cable meets specifications. Let S and L
be the events that the cable is too small and too large, respectively. Then