Transcript MAT 1000 - Wayne State University

MAT 1000
Mathematics in Today's World
Last Time
We talked about Cryptography
Today
We will talk about probability.
There are four rules that govern probabilities.
One good way to analyze simple probabilities is to make a
list of everything that can happen—this is called a sample
space.
A probability model is the sample space with probabilities
for each outcome.
Random Phenomena
A phenomenon is random if each individual occurrence is
uncertain, but there is a pattern over many occurences.
Examples of random phenomena:
•
•
Tossing a coin, rolling a die, drawing a card.
The gender of a baby.
Random Phenomena
To describe random phenomena, start by giving a list of
everything that could happen. Each distinct thing that could
happen is called an outcome.
If we toss a coin, there are two outcomes: heads or tails.
We can’t predict which will happen, but in the long run 50%
of the tosses are heads and 50% are tails.
Random Phenomena
If we roll a (six-sided) die, there are six possible outcomes:
We can’t predict in advance which outcome will occur.
In the long run each outcome will occur 1/6th of the time.
Random Phenomena
We don’t know in advance if a baby will be a boy or a girl.
In the long run it turns out that about 49% of newborns are
girls, and 51% are boys.
Note that even though there are two outcomes, they don’t
have to occur in equal proportions.
Random Phenomena
To describe random phenomena, start by giving a list of
everything that could happen. Each distinct thing that could
happen is called an outcome.
Each outcome of a random phenomenon has a probability.
The probability of an outcome is the proportion of times it
occurs over many trials.
Probabilities are proportions, so we can express them as
percents (50%), decimals (0.5), or fractions (1/2),
whichever is most convenient. I will use all of these.
Random Phenomena
We can restate our earlier observations in terms of
probabilities:
When we flip a coin the probability of getting tails is 50%.
When we roll a die, the probability of getting a 4 is 1/6
The probability that a newborn baby is a girl is about 49%
Sample Space
When there are not too many outcomes, one good way to
analyze a random phenomenon is to make a list of every
outcome.
The set of all of the outcomes of a random phenomenon is
called the sample space.
For tossing a coin the sample space is
{heads, tails}
For rolling a die the sample space is
{1, 2, 3, 4, 5, 6}
Random Phenomena
We use the term event to describe a subset of the sample
space.
For example, when we roll a die, we can look for the event:
“rolling a odd number.”
There are three outcomes in this event: rolling a 1, a 3, or a
5.
As a set, this event is
{1, 3, 5}
The probability of this event is
1
2
Random Phenomena
Another event is “rolling a 6”
There is only one outcome in this event.
As a set, this it is
{6}
The probability of this event is
1
6
Probability models
The sample space tells us all of the possible outcomes.
If we include the probability of each outcome we get what is
called a probability model.
Example
Here is the probability model for tossing a coin:
Heads
0.50
Tails
0.50
Probability models
Here is a probability model for the gender of a newborn:
Boy
0.51
Girl
0.49
Here is a probability model for tossing a coin:
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
The top row is a list of the outcomes. The bottom row is the
probability of each.
Probability Rules
There are four rules which govern probabilities.
Any probability is a number between 0 and 1.
A probability of an event is the proportion of times that it
can be expected to occur. The larger the probability, the
more likely it is.
Probability Rules
If the probability of an event is 1, it will always happen for
sure.
If the probability of an event is 0, it never happens.
Note that neither of these are really random!
Probability Rules
The probability that an event does not occur is 1
minus the probability that the event does occur.
What is the probability that we roll a die and do not get
a 6?
The probability of getting a 6 is
1
.
6
By the rule, the probability of not getting a 6 should be
1 5
1− =
6 6
Probability Rules
If two events have no outcomes in common, they are
said to be disjoint. The probability that one or the
other of two disjoint events occurs is the sum of their
individual probabilities.
The two events “rolling an odd number” and “rolling a 6”
are the sets
{1, 3, 5}
{6}
These sets are disjoint (nothing in common).
Probability Rules
By the rule, the probability that we roll either an odd
number or a 6 is the sum of their individual probabilities:
1 1
+
2 6
To add these fractions we can use a common
denominator of 6
3 1 4
+ =
6 6 6
This can be reduced
4 2
=
6 3
Probability Rules
All possible outcomes together must have
probability 1.
Because some outcome must occur on every trial, the
sum of the probabilities for all possible outcomes must
be exactly one.
Probability models
We can use this last rule to fill in missing information in a
probability model.
If you draw an M&M candy at random from the bag, it will
have one of six colors. Here is a probability model:
Brown
Red
Yellow
Green
Orange
Blue
0.30
0.20
0.20
0.10
0.10
?
What is the probability of randomly drawing a blue M&M?
The probabilities of all the outcomes must add up to 1.
So
0.30 + 0.20 + 0.20 + 0.10 + 0.10 + Blue = 1
The probability the M&M is blue is 0.10
Independent Events
If the outcome of one event has no impact on the outcome of
the other, we say the two events are independent.
Each roll of a die is independent from any other roll. Why?
Because the outcome of one roll does not affect the next roll.
The weather one day is not independent from the weather the
next day. Why?
Because weather moves in systems: if it rains one day, the
probability of it raining the next day is increased.
Independent Events
The most important fact about independent events is that the
probability that both happen is the product of the individual
probabilities.
The probability that I get heads on a coin toss is 1/2. The
probability that I get a 1 when I roll a die is 1/6.
These are independent events.
So if I toss a coin then roll a die, the probability that I get
heads on the coin and a 1 on the die is the product:
1 1
1
⋅ =
2 6 12
Probability models
We are now going to put all of this information together to start
constructing more complicated probability models.
Remember a probability model is a list of outcomes and their
probabilities.
Probability models
If we toss two coins, what are the possible outcomes?
We can get heads on both, tails on both, or heads on one and
tails on the other.
But be careful, there are actually four outcomes. Why?
This is easiest to see if you imagine the coins being different,
say one is a quarter and one is a dime:
Quarter
Dime
H
H
H
T
T
H
T
T
Probability models
Now we have a sample space with four outcomes. Let’s make
a probability model.
What is the probability of getting heads on each coin? The
coin tosses are independent, so
probability of heads on the quarter and heads on the dime =
(probability of heads on the quarter) X (probability of heads on the dime)
The probability of getting heads on both is 0.50 0.50 = 0.25
In a similar way the probability of each of the other three
outcomes is also 0.25
Probability models
So the probability model for tossing two coins is:
H,H
0.25
T,H
0.25
H,T
0.25
T,T
0.25
What is the probability of tossing two coins and getting one
head and one tail?
This is an event, containing two outcomes: T,H and H,T
So the probability of one head and one tail is
0.25 + 0.25 = 0.50
Probability models
Here’s a related example: let’s find a probability model for
rolling two dice. First we should make a sample space:
Again we distinguish getting a 3 on the first die and a 4 on the
second from getting a 4 on the first and a 3 on the second.
Probability models
It is useful to write outcomes here as pairs of numbers.
So (3,4) indicates a 3 on the first die and a 4 on the second.
As I said on the last slide: the outcome (3,4) is not the same
as the outcome (4,3).
Now we know the sample space. What are the probabilities of
each outcome?
Probability models
It turns out that each outcome has the same probability:
How many outcomes are there? 36.
They all have the same probability, so each one has
probability 1/36.
Probability models
Now, suppose we roll 2 dice and then add the numbers we
get.
So if we get (2,5), the outcome is the sum of these
2+5=7
What is the sample space?
The possible outcomes (sums of two numbers between 1 and
6) are
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Probability models
What’s the probability of each of these outcomes?
Look back at the sample space for rolling two dice:
We get a sum of 2 only from the outcome (1,1). The
1
probability is
36
There are two ways to get a sum of 3: (1,2) and (2,1). The
1
1
2
1
probability is + = =
36
36
36
18
Probability models
How many ways are there to get a sum of 4?
There are three ways: (1,3), (2,2), or (3,1). The probability of
getting a sum of 4 is:
1
1
1
3
1
+
+
=
=
36 36 36 36 12
Continuing in this way, we get the following probability model:
Outcome
Probability
2
3
4
5
6
7
8
9
10
11
12
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
=
1
1
=
18
12
=
1
9
=
1
6
=
1
9
=
1
1
=
12
18
Probability models
It can be useful to represent probability models using
probability histograms. The horizontal scale gives the
outcomes, and the probability of each outcome determines the
height of the bar: