Transcript P(n,r)

If you roll a 2 you will
get no homework. What
are your chances of
getting no homework?
First we need to find out
what could happen when we
roll the dice. Find all of the
possible things that could
happen . These are called
the outcomes.
The specific outcome we are
looking for is 2. This is
called the event. In this
experiment we have 1
chance of getting a 2
The outcomes are1,1:1,2:1,3:1,4:1,5:1,6 If we continue this
pattern we will have 36 possible outcomes.
In words, the probability of an event is the ratio of the
number of ways an event can occur to the number of
possible outcomes.
In symbols P(event) =
The number of ways the
event can occur
Number of possible
outcomes
In our experiment
P(2) = 1
36
So the probability of getting no
homework is
one in thirty-six
Course 1 wb 14-1
Main Dish
Desert
Drinks
The cafeteria is offering the above items for lunch.
For $ 3 you get one item from each section.
If the menu stays the same how many different
meals could you eat?
One way to find out is by listing all of the possible
outcomes which is called the Sample space.
By following down the branches
you can determine all of the
possible choices or outcomes and
see the entire sample space.
Course 3 wb 13-1
A tree diagram can be made horizontally and by using words or letters to
represent the choices.
Main Dish
HB
P
Drinks
Desert
PIE
HD
PIE
PIE
P
IC
IC
M
IT
M
HB,PIE,M
IT
HB,PIE,IT
HB
IC
HD
Course 2 wb13-1
Outcomes
M
HB,IC,M
IT
HB,IC,IT
M
P,PIE,M
IT
P,PIE,IT
M
P,IC,M
IT
P,IC,IT
M
HD,PIE,M
PIE
IT
HD,PIE,IT
IC
M
HD,IC,M
IT
HD,IC,IT
When looking for just the total number of outcomes, using the
Fundamental Counting Principle is quick and easy.
Main dishes
3
Drinks
Deserts
x
2
x
2
= 12
The lunch problem had 3 groups of choices. Multiplying the number
of choices in each group provides us with the number of outcomes.
There are 12 different lunch combinations.
Course 2 wb13-2 course 3 13-1
Theoretical probability is a mathematical
computation using the ratio
The number of ways the event can occur
Number of possible outcomes
This type of probability can allow us to predict what could happen.
But what actually happens may be different.
If a coin is flipped we can get heads or tails.
Our theoretical probability of heads is
1/2.To find out how many heads we could
expect in ten trials (flips) multiply 1/2 by
10 (5 heads).
Roll the die
Flip the coin
Spin the spinner
1
2
4
3
To find P(1,3,H) use the fundamental counting principle
P(1)
1/6
P(3)
x
1/4
P(H)
x
P(1,3,H) = 1/48
This is the theoretical probability.
Course 3 13-5 course 2 13-6
1/2
The previous experiment is an example of an independent event. (The roll of
the die had no impact on the the spinner or the coin)
Experiment: Pick a ball from box 1 and place it in box 2. Then pick
a ball from box 2. What is P(B)?
In box 1 a black or white ball can be picked.What happens in box 2 is
dependent on what happens in box 1 since the # of each color will
change depending on what is picked from box 1. This experiment is a
Dependent Event.
A tree diagram can be very helpful in working with Dependent Events.
First, draw the diagram and list the P on each
branch. Next label the possible outcomes.
2/4
2/3
W
2/4
W
B
WB
2/3 x 2/4 = 4 /12
W
BW
1/3 x 1/4 = 1/12
BB
1/3 x 3/4 = 3/12
1/4
1/3
B
3/4
OUTCOMES
P
WW 2/3 x 2/4 = 4/12
B
Then multiply along the branches to get the
P of each outcome. Finally, add the events that end in B:4/12 +3/12 =7/12
Theoretical probability is based on mathematical principles.
Experimental Probability is an estimated probability based on the relative
frequency of positive outcomes occurring during an experiment.
Experimental probability does not always coincide with
theoretical probability. Many organizations use experimental
probability to make predictions or forecasts of future trends.
Surveys are often used to obtain the data for the basis of the
experimental probability.
Course 2 wb13-3
Below are the results of a survey of the 6th and 7th grade students in
Norwood. If there are 325 - 6th graders and 350 - 7th graders in the
Valley, about how many of each class will prefer KROC?
KROC 92.3
6TH GRADERS
24
7TH GRADERS
26
WPLJ 95.5
30
32
Z 100
KTU 103.5
13
43
15
38
Q 104.3
21
19
The survey represents a sample population. We can use this data to
obtain an experimental probability. However, it is only useful when
applied to a similar population. (For example using this data to predict
what station their parents might prefer would not be useful because the
population “parents” is not similar to the population “6th and 7th
graders”)
KROC 92.3
6TH GRADERS
24
7TH GRADERS
26
WPLJ 95.5
30
32
Z 100
KTU 103.5
13
43
15
38
Q 104.3
21
19
First, get a total of each population 6th = 131 7th = 130( these will
be our denominators)
Next, find the event we are trying to find (KROC92.3
6th = 24
24
26
131
130
24
131
7th = 26). These will serve as our numerators.
x 325 = 60
Then multiply the ratios by the total
number of the new population.
26
x 350 = 70
130
Course 3 wb 13-7 course 2wb13-5
A sky diver is going to land in the school field. If it is equally likely to
land on any part of the field what is the probability he will land in a
circle in the middle of the field?
400 feet
125 feet
Area of circle
P(circle) = Area of rectangle
42 feet
A = Pi r * r
=3.14 *21 * 21
P(circle) = 1385
= 2.8%
=1385 sq ft.
A=l*w
50,000
= 400 *125
= 50,000 sq ft
Course 1 wb14-4
Belgium
The flags of Germany and Italy are made
of black, yellow, red, green and white
stripes stripes.
Italy
How many different flags can be made of
these colors using only vertical stripes?
For the 1st color we have 5 choices. After the 1st
stripe is used we have 4 choices for the next stripe.
Then we have 3 colors left for the 3rd stripe.
To solve we can use the counting principle 5 x 4 x 3 = 60
different flags.
This problem is an example of a permutation. A permutation is an
arrangement of objects in which order is important.
Notice that black,yellow
and red is not the same flag
as red, yellow and black.
In our problem we have 5 choices to be taken or used 3 at a time.
The permutation formula is P(n,r) n is the number of items or
choices and r is how many
are used or taken at a time
If we were making a flag with 4 stripes our formula would be:
P(5,4) 5 colors(choices)
taken 4 at a time (4 stripes)
5 x 4 x 3 x 2 = 120
different flags.
Course 2 wb 13-7
course 3 wb 13-2
How many 4 digit numbers can be made from 3, 5 ,7 ,9?
This is a permutation of 4 things taken 4 at a time P(4,4)
To solve we multiply 4 x 3 x 2 x 1 Another way of writing this is 4!
This is read 4 factorial.
6!
In words n! is the product of all of the
counting numbers starting with n and
counting backwards to 1.
In this problem we have 6 x 5 x 4 x3 x2 x 1
4!
4x3x2x1
In this problem we can simplify.
Our answer is 6 x 5 = 30
Joe, Sam, Tom and Bob are all guards for the basketball team.
In how many ways can the coach choose 2 starting guards?
In this problem
Joe and Tom are
the same as Tom
and Joe.
So the order is not
important.
We ca make a list:JS, JT, JB,ST, SB, TB.
These are all of the different combinations since JS and SJ are
the same.
So we have six different combinations.
This problem is an example of a Combination. A combination is
an arrangement in which the order is not important.
In our problem whether we list Tom and Joe or Joe and
Tom, it is still the same combination of players.
The format for combinations is C(n,r) which in words means
the number of combinations of n things taken r at a time.
Mathematically we write it:
P(n,r)
C(n,r) =
r!
The basketball problem can be solved like this:
C(4,2) = P(4,2)
2!
=4x3
2x1
= 12
2
= 6 combinations.
Course 3 wb 13-3 Course 2 wb 13-8