Wednesday, March 28, 2007

10.7: Direction Angles and Direction Cosines

Direction Angles and Direction Cosines

OBJECTIVE: given a vector, find its direction angles and direction cosines and vice versa

The graph shows a vector v and its direction angles:




the first three letters of the Greek alphabet alpha, beta, and, gamma are used for the three direction angles. like x, y, and z, the letters come in alphabetical order, corresponding to the three axes.

DEFINITIONS: Direction angles and Direction cosines:



the direction angles of a position vector are:


alpha, from the x-axis to the vector
beta, from the y-axis to the vector
gamma, from the z-axis to the vector
the direction cosines of a position vector are the cosines of the direction angles:


c1 = cos (alpha)
c2 = cos (beta)
c3 = cos (gamma)


PROPERTIES: Direction Cosines:


Pythagorean Property of Direction Cosines:
if alpha, beta, and gamma are the direction angles of a position vector and c1 = cos (alpha), c2 = cos (beta), and c3 = cos (gamma) are the direction cosines, then
cos squared (alpha) + cos squared (beta) + cos squared (gamma) = 1
or
c1 squared + c2 squared + c3 squared = 1


Unit Vector Property of Direction cosines:
vector u = c1 vector i + c2 vector j + c3 vector k is a unit vector in the direction of the given vector.




Example Problem:

find the direction cosines and the direction angles for vector v = 3 vector i + 7 vector j + 5 vector k

Solution:

find the dot products of vector v dot vector i, vector v dot vector j, vector v dot vector k. then use these to find the angles.

vector v dot vector i = (3 vector i + 7 vector j + 5 vector k) (1 vector i + 0 vector j + 0 vector k) = 3, which is the coefficient of vector i in vector v

vector v dot vector i = (3 vector i + 7 vector j + 5 vector k) (0 vector i + 1 vector j + 0 vector k) = 7, which is the coefficient of vector j in vector v

vector v dot vector i = (3 vector i + 7 vector j + 5 vector k) (0 vector i + 0 vector j + 1 vector k) = 5, which is the coefficient of vector k in vector v

absolute value of vector v = square root of 3 squared + 7 sqaured + 5 squared = squared root of 83

abolute value of vector i = abolute value of vector j = abolute value of vector k = 1
vectors i, j, and k are unit vectors

square root of 83(1) cos alpha = 3 ==> cos alpha = 3/square root of 83 ==> alpha = 70.774 degrees

square root of 83(1) cos beta = 7 ==> cos beta = 7/square root of 83 ==> beta = 39.794 degrees

square root of 83(1) cos gamma = 5 ==> cos gamma = 5/square root of 83 ==> gamma = 56.713 degrees






Kaori, you're up next! Good luck!


Who is this country singer????

10-6:Vectors Product of Two Vectors

Cross Products
  • The Cross product of two vectors a x b is perpendicular to the plane containing vectors a and b.
  • The magnitude of a x b- the absolute value of vector a multiplied by the absolute value of vector b multiplied by sin(pheta).
  • The direction of a x b is determined by right hand rule.
Right hand rule- Put the fingers of your right hand so they curl in the shortest direction from the first vector to the second vector. The cross product is in the same direction your thumb points.
  • Thumb points in-rotates from vector b to a and b x a
  • Thumb points out-rotates from vector a to b-a x b
Cross products of the unit coordinate vector

i x i=0 i x j=k j x i=-k

j x j=0 j x k=i k x j=-i

K x K=0 k x i=j I x k=-j

Example of Cross Product

Find vectors a x b
a= 4i+5j+9k and b=11i+5j+10k
a x b= 44 i x i + 20 i x j + 40 i x k
+ 55 j x i+ 25 j x j + 50 j x K
+ 99 k x i + 45 k x j + 90 k x k

2ok-40j
-55 k + 50 i
99 j - 45 i

=5i+59j-35k

In determinants-Example 1

(i J K )
( 4 5 9 )
(11 5 10)

i
(5 9 ) -j (4 9) +K (4 5)
(5 10) (11 10) (11 5)

I(5)(10)-(5)(9) - j(4)(10)-(11)(9)+ k(4)(5)-(11)(5)

=5i+59j-35k

Geometrical cross products and meaning of a x b

a
rea of a parallelogram have a and b as adjacent sides= absolute value (a x b)

area of triangle have a and b as adjacent sides= 1/2 absolute value (a x b)

Example

Find the are of the triangle with vertices's P1(-5, 5, 5), P2(-3, 2, 7), P3(1, 12, 6)

P1P2 x P1P3= -17i + 10 j + 32k

Area= 1/2 (-17i + 10 j + 32k)= 1/2 square root of (-17)^2 + 10^2 + 32^2

= 37.5898


Kaori, your up next good luck

Additional sources-
http://en.wikipedia.org/wiki/Cross_product- This website page gives one great detail on the subject of Cross Products.

Personal touch- Kobe Bryant had a great week in basketball, scoring 50 points or more in 5 games straight.

Here a video of his career, check it out-http://www.youtube.com/watch?v=mGbquGyTW34

Monday, March 26, 2007

Lesson 10-5: Planes In Space

In class today we learned how to find the equation of a plane using a point and perpendicular vector contained on the plane. The equation for a plane in space is:
Ax + By + Cz =D
A, B, and C: coefficients of the components of a normal vector
D: value determined after substituting the coordinates of a given point



First, let’s review the equation using a problem…

The figure below shows a plane in space with a vector , and a point P (4, 6, 8) both contained on the plane. How do we find the equation of the plane?


NOTE: To determine this equation, we have to find an equation of the plane relating x, y, and z where the vector is perpendicular to all points

Step One: Using the given vector equation and point, find a new point contained on the plane.

vector:

point a:
(4, 6, 8)
new point b:
(x, y, z)

Step Two: Now that we have two points, we can create a line connecting the point given to us (4,6,8) to point (x,y,z). As we learned from the mini-quiz, multiplying these two points together gives us a dot product of 0, which is equal to the cos(90°).



Step Three: Next we need to distribute the equation of our vector:

to the equation relating our two points (v).



Step Four: Then, we multiply both sets of equations and set the result equal to zero.



Step Five: With this last equation, we discovered that the coefficients were the same as those of the perpendicular vector equation:



Thus, to find the equation of the plane more efficiently, we can use the vector coefficients and replace them with A, B, and C. To find D, we simply distribute the vector coefficients to the corresponding point.

We distribute the coefficients 12, 3, and 14 to the corresponding x, y, and z values of the ordered pair (4, 6, 8). Thus, 12 x 4, 3 x 6, and 14 x 8. After adding up these numbers, we get 178. Thus, our final equation is:

178 = 12x + 3y + 14z

Helpful Notes:

  • If you are given the equation for a plane and need to find the equation for the perpendicular vector, remember the coefficients are the same! All you have to do is replace x, y, z, with i, j, k !
  • If you are given the equation for a plane and need to find a coordinate for your point on the plane, just plug in the coordinates for x, y, and z, and use basic algebra to solve!

For More Info…
Although this website seems a little more complicated, the animation helps me visualize exactly what the equation and subsequent steps are for solving:
http://college.hmco.com/mathematics/larson/precalculus_limits/1e2/ins_resources/ap.html

Fun Stuff…
This is from my favorite comic site, toothpastefordinner.com. (its my favorite comic for the month):
http://www.toothpastefordinner.com/012207/latest-poll-results-what.gif

Annie, you are up next! Good luck!

Thursday, March 22, 2007

Friday's Quiz Topics

Here’s a list of topics for Friday’s quiz:

Precalculus Quiz 10.1-4 Topics
Draw a three-dimensional vector and its “box”
Unit vectors: i, j, k
Adding vectors visually
Subtracting vectors visually
Dot/Scalar/Inner Product calculations – both formulas
Projection of a vector onto another vector – visual
Calculate vector projection of one vector onto another vector
Magnitude of a vector – calculation
Dot product – calculation
Calculate the angle between two vectors
Scalar projection calculation – two methods
Calculation of a unit vector
Vector projection calculation
Three-dimensional resultant forces
Angle between two three-dimensional vectors
Three-dimensional position vectors
Position vector to a point along a displacement vector


That’s it! The format’s the same as always – ½ Non-calculator, ½ Calculator. There are 10 questions on the Non-calculator portion and 12 questions on the Calculator section. I’ll be around briefly after school on Thursday, and in early on Friday morning.

I don’t know whether my life has been a success or a failure. But not having any anxiety about becoming one instead of the other, and just taking things as they came a long, I’ve had a lot of extra time to enjoy life.
—COMEDIAN HARPO MARX

Wednesday, March 21, 2007

10-4: Scalar Products and Projections of Vectors

10-4 Scalar Products and Projections of Vectors



The Objective: In a situation where you are given two vectors, find their dot product. Use the result to find the angle between the vectors and the projection of one vector on the other.

Basically this chapter is dealing with multiplying vectors together using various methods and answering various questions.

*An important thing to remember* is that dot products, scalar products, and inner products are all the same thing.

The equation for the DOT PRODUCT of vector a and vector b is vector "a" times vector "b" or = a dot b (both are vectors)

If you translate the vectors so that they are tail-to-tail you find the dot product by multiplying the magnitudes of the vectors and the cosine of the angle between them



For example (vectors "v" and "u"):
















The definition of a dot product is vector "a" times vector "b" equals the magnitude of vector "a" times the magnitude of vector "b" times the cosine of the included angle.



Technique for the computation of dot product:

vector a= X1i + Y1j + Z1k
and
vector b= X2i + Y2j + Z2k
then

a dot b = X1X2 + Y1Y2 + Z1Z2

Maybe it will help to clarify in verbal terms: The dot product of two three-dimensional vectors equals the sum of the respective products of the coefficients for the i, j, and k unit vectors.


Example


Find the dot product of vectors a and b.


vector a= 3i - 4j + 12k
vector b= 2i + 6j - 3k


a dot b = 3(2) + (-4 )( 6) + (12 ) (-3) = -54


To find the measure of the angle theta between vectors x and y you would take:


a dot b = -54



































(7 ) (13) ( cos theta)= -54

cos theta= -54/ (7 ) (13 )

cos^-1(-.5934)= 126.399 degrees (theta)



The other portion of this section deals with Projections of Vectors

The component is called vector projection of vector a on vector b.




An example of this would be light rays shining on two vectors. The rays would be perpendicular to the bottom vector made up of two black arrows (forming a right angle).






Let vector "p" be the vector projection of vector "a" on vector "b".

The formula used for this projection would be:

vector p=

There are two types of scalar projections, acute and obtuse.




Acute: an angle less than ninety degrees (x<90)















Obtuse: an angle greater than ninety degrees (x>90)













Finally, the scalar projection of a on b is



p= magnitude (a) cos theta



and the vector projection of vector a on b is



vector p= (p ) (unit vector "u")



If you need help with any of this material go to http://distance-ed.math.tamu.edu/Math640/chapter2/node4.html it is rather helpful in general.



Take this quiz to find out what kind of gnome you are!



http://quizfarm.com/test.php?q_id=226384





Good luck Celene you are next!






Tuesday, March 20, 2007

Lesson 10-3-Vectors in Space

The Overview of the lesson



This Image on the left is referred to as an Octant.

Octant-The xy-plane, yz-plane, and xz-plane divided into 8 regions.

The region in which all the three variables are positive is called first octant.

The X axis- i variable, the Y axis- j variable, and the z axis- k variable. The variables are used for unit vectors in the x-, y-, and z- direction respectively.





To finding the magnitude is similar to finiding the 2-dimensional magnitude.

ex-
Example of this lesson


Example #1- Find the length of vector P. Vector P=4i+5j+9k

Magnitude (length)- square root of (4^2 + 5^2 + 9^2)

Magnitude=11.045

Example #2- Find the displacement vector of point A(9,3,14) to point B(4,11,5).

One always subtracts from head to tail.

(4i+11j+5k) - (9i+3j+14k)= -5i+8j-9k=Displacement vector from A to B.

Example #3- Find the positon vector to the point 70% of the way from point A(9,3,14) to point B(4,11,5).

Position vector is A+.7(AB)

9i+3j+14k + .7(-5i+3j-9k)= 5.5i+5.1j+7.7k

Additional source

Additional Source that will help is found online on Wikipedia. Type in the search vectors and one will find online assitance.

Allison your next up-good luck

I hope my UCLA Bruins win it all this March in the tornument. This is a picture that examplifies UCLA Basketball.






Monday, March 19, 2007

ARASH YOU'RE NEXT!

Section 10-2: Two Dimensional Vector Practice


Hey kids in this lesson we reviewed our vectors. We learned a bunch of new vocab which I will give you further down. Hope you don't find this too frustrating. Also, I couldn't figure out how to get the arrows above the letters, so if you think there should be an arrow above it, it's because there should be.


Vector Definitions and Properties
Vector Quantity - has both magnitude and direction (eg. - force, velocity, displacement)
Scalar - quantity but no direction (eg. - weight, height)
Vector - directed line segment that represents a vector quantity (v w/ an arrow above it)
Tail - where the vector begins
Head - where the vector ends
Magnitude - (absolute value is used) the length of a vector
absolute value(v), If v = xi = xj, Then absolute value(v) = square root (x² + y²)
Unit Vector - absolute value(u) in the direction of v is a vector that is one unit long in the same direction as v. So u = v/abs(v)* - divide the vector by it's length
Vectors are equal if they have the same magnitude and direction.
The opposite of a vector is the same length in the opposite direction (-v)
Position Vector - v = xi + yj; Starts at the origin and the end point is (x,y)
When adding two vectors the resultant vector falls from the tail of the 1st vector to the tip of the 2nd. (Start of the 1st to the end of the 2nd)
* abs() = absolute value

Example Problem 1
If a = 4i + 8j and b = 5i - 3j
Find a+b => Add the two equations => 9i -5j
Find a-b => Subtract the two equations=> -i + 11j
Find -a => -4i - 8j
Find 2a + 4b => First multiply a by 5 and b by 4 => 8i +16j and 20i - 12 j => then add => 28i +4j
When doing problems like this make sure you understand that the abs(a) + abs(b) does NOT equal the abs(a + b)
Also, when finding the angle for the vector, use the inverse tangent.

Example Problem 2
Given point A (5, 10) and B (8, 20)
Find vector AB (Pointing from A to B)
to do this, look at the given points and subtract. AB = 3i - 10j
Find the position vector of the point 3/4 of the way from A to B.
To do this, it is easier to draw the postion vector to point A, then determine what is 3/4 of AB. Once that is found, add the two vectors.
A = 5i + 10j; 3/4 AB = 3/4(3i - 10j) = (9/4)i - (30/4)j
The answer would then be 5i+10j + (9/4)i - (30/4)j
This equals => 7.25i + 2.5j

Now in honor of March Madness, and USC beating Texas, here is a picture of the Trojan basketball team.

Tuesday, March 13, 2007

Questions and Answers from Chapter 9 Review

Here They Are...


The Questions and Answers for Factorial Feud

1. What are mutually exclusive events?

Events in which the occurrence of one event excludes the possibility that the other will occur.

2. A box of pencils has 35 lead pencils and 240 colored pencils. How many ways can you pull out a colored pencil and then a lead pencil?

8400 ways


3. 6 students must form a line. In how many ways can 6 line up from a group of 20?

27907200 ways

4. What is a permutation?

An arrangement in a definite order of some or all of the elements in a given set.

5. What are complementary events?

Events that complete all possibilities of a random experiment . the probability of any of the complementary events occurring is 1.

6. 20 people enter a competition to make it in America’s Best Barbershop Quartet. If only 4 can be selected, how many possible quartets are there?

4845

7. How many different circular permutations could be made with the letters “MY HOUSE”?

720

8. What is the probability of selecting an A, N, or E from the words “BANANA PEELS”?

7/11 or 63.636%

9. How many ways could 14 friends sit around a round table?

6227020800 ways

10. What is a combination?

A combination of elements in a set is a subset of those elements, without regard to the order in which the elements are arranged.

11. Without using your calculator, determine the value of 12C7

792

12. There are 14 mechanical pencils on a desk. 8 are blue, and six are green. If you choose six of them, what is the probability of 3 being blue, 3 being green? 6 being green?

160/429 or 37.3%, 1/3003

13. What does the following mean: A ∩ B?

The set of outcomes in sets A and B/ the intersection of A and B.

14. What does the following mean: A ∪ B

The union of A and B- the set of all outcomes in A or B.

15. If you are planting 2 flowers, and there is a 70% chance of the first living, and a 4% of the second living, what is the probability of both living? Neither living? The second living but not the first?

2.8%, 28.2%, 1.2%

16. Four fleas are living on a dog’s back. If the dog gets a bath, the first flea has a 20% chance of surving, the second has a 43% chance, the third has a 12% chance, and the fourth has an 87% chance. What is the probability of all surviving? None surviving? Just the third and fourth living? Only the 3rd? only the 4th?

.898%, 5.217%, 4.761%, .711%, 34.911%

17. Two dice are rolled (one pink and one purple). Find the probability that: (do these quickly-you should be able to)
a. The total is 10.
b. The total is 7.
c. The total is 2.
d. The numbers are 2 and 5
e. The pink die shows 2 and the purple die shows 5
f. The pink die shows 2 or the purple die shows 5.
1/12, 1/6, 1/36, 1/18, 1/36, 11/36

18. What is a random experiment?
The act of doing something
-
19. What is a trial?
Each time you do something

20. What is another term for outcome?
Simple event

21. What is the term for all the possible outcomes?
Sample space

22. A card is drawn from a 52 card deck.
a. How many outcomes are in the sample space?
b. How many outcomes in the event that the card is a face card?
c. What is the probability that the card is a face card?
d. What is the probability that the card is red?
e. What is the probability that the card is a king?
f. What is the probability that the card is the king of hearts?
g. What is the probability that the card is in the deck?
52, 12, 12/52 or 3/13, _, 1/13, 1/52, 1

23. 8% of men have a peg leg. Suppose 20 men are selected at random at let P(x) be the probability that x of the men have a peg leg.
a. Find the binomial probability equation for this scenario and find the values for x= 0,1,2,3.
P(x)=
.18869, .32816, .27109, .14143

24. Bobby’s batting average is .3. He comes to bat 5 times during a game.
Calculate the probabilities that he got 0, 1, 2, 3, 4, and 5 hits and what is the mathematical expectation of hits for this game?

.168, .360, .309, .132, .028, .0024
Mathematical expectation is 1.5

Thursday's Test Topics

Here’s a list of topics for Thursday’s test:

Precalculus Chapter 9 Test Topics:
Calculate simple permutations/combinations
Know the difference between permutations and combinations
Calculate Probabilities: And vs. Or (For independent/non-independent and exclusive/overlapping events)
Complementary Probabilities
Binomial Experiments (x successes out of n trials)
Binomial Expansions
Mathematical Expectations

That’s it! The format is as expected – ½ calculator, ½ non-calculator. 10 questions on the Non-calculator portion, 17 on the Calculator portion. I’ll be in my classroom for a short period on Wednesday afternoon and before school on Thursday. I’ll try to be available online later on Wednesday evening. If you have specific questions Wednesday night, email me!

I’ve posted a sample test on the class website which I’ll distribute in class tomorrow.

See you in class!

I like nonsense, it wakes up the brain cells. Fantasy is a necessary ingredient in living, It's a way of looking at life through the wrong end of a telescope. Which is what I do, And that enables you to laugh at life's realities.
- Dr. Seuss


Saturday, March 10, 2007

9-8 Mathematical Expectation

Lesson 9-8: Mathematical Expectations

In this lesson, we will learn about how you can calculate an expected value based on outcomes for each event in a random experiment. Basically, we will put our knowledge of probability to good use.

The example the book gives is of a carnival game of rolling a die.
If you...
roll a 6-- you win 100 points
roll a 2 or 4–- you win 10 points
roll an odd number–- lose 50 points

So in chart form,

Number on die ------Points you win
1 --------------------(-50)
2--------------------(10)
3 --------------------(-50)
4 --------------------(10)
5 --------------------(-50)
6 --------------------(100)

Strategy No. 1: To find the average point winning from the rolling-a-die-game...
You add up all the possible point outcomes and divide by the number of rolls!

(-50+10-50+10-50+100)/(6rolls) It simplifies too..
-30points/6 rollsor -5points/roll


What does it mean???---This answer means that you will on average lose 5 points per roll of the die.

Strategy No. 2

(-50 + 10 + -50 + 10 + -50 + 100) /divided by 6 rolls.
--- (The different outcomes of pts are in numerator, and the number of rolls/trials is in denominator)

(3*(-50) + 2*(10) + 1*(100)) /divided by 6 rolls --- you can "combine like terms"

3/6 * (-50) + 2/6 *(10) + 1/6 * (100) ---combine into individual fractions so that you can see what the probabilities for each outcome is!!!

probability of losing 50 pts is 3/6
probability of winning 10 pts is 2/6
probability of winning 100 pts is 1/6

Strategy No. 3 is the Algebraic Formula:

Mathematical Expectation = (Probability of Event A) * (Outcome of Event A) + (probability of event B) * (outcome of event B) +...

In the die-rolling example:avg winning = [(probability of rolling a 6) *(100 points)] + [(probability of rolling an odd number) * (-50 points) ] + [(probability of rolling a 2 or 4) * (10 points)]

PROBLEM #2:
The 2nd example that the book gives is more difficult because the probability involves "binomial probability distribution" from section 9-7...which means you have to use the equation from pg. 387 of the text book...

P(x) = nCx * a^n-x * b^x

b is the probability that the event occurs
a is the probability that the event does not occur
x is the number of times the event occurs in n repetitions

NOTE: use binomial probability distribution when there are only 2 possible outcomes...like getting a basketball shot---you either get it in or not. Whereas, you wouldn't use this with the die problem because there is an equal probability of rolling each of the 6 numbers on the die.

Basketball Toss Problem: You get 3 tries to shoot basketball hoops.

Number of Baskets------- Points (outcome)
0-------------------------(-50)
1 -------------------------(-45)
2 -------------------------(10)
3 -------------------------(200)

Probability of making the shot = 30% or .3
Probability of not making the shot = 70% or .7

To Solve!!!Now, find the probability of making x baskets...USE binomial probability distribution equation!!!



P(0) = 3C0 * .7^3 * .3^0 = .343 probability of getting 0 shots

P(1) = 3C1 * .7^2 * .3^1 =.441 prob. of getting 1 shot

P(2) = 3C2 * .7^1 * .3^2 = .189 prob. of getting 2 shots

P(3) = 3C3 * .7^0 * .3^3 = .027 prob. of getting 3 shots

(**note: the probabilities of all the outcomes add up to 1)


Now, just plug it into the Mathematical Expectation Equation!
mathematical expectation (E) = (prob. of 1st event * outcome of 1st event) + ...2nd...3rd....

E = .343 * (-50) + .441 * (-45) + .189 * (10) + .027 * (200)

E= about -30 points per game

ALL DONe!

Sites for more information on mathematical expectation:
http://www.wiu.edu/users/mfmk/Math101/Probability/PExpect.html
http://engrwww.usask.ca/classes/GE/210/CE%20225%20Webpage%20files/LECTURES/Lecture%208/Updated%20Lecture%208.pdf


Personalized Part: I love Miró. In my 3rd grade class with Mrs. Schroeder, we learned about Miró and all tried to copy his style...



























Reminder!! Nick, you are NEXT!

Tuesday, March 06, 2007

Chapter 9 Section 6: Properties of Probability

Hey everybody! Since I'm posting this blog before Mr. French has lectured on the section, please excuse any mistakes. I will be sure to correct anything that is wrong as soon as I'm aware of it.

Intersection of Events
AND:

1.) p(A and B)=p(A) x p(B) Independent events
2.) p(A and B)=p(A) x p (B/A) Non-independent events
~Note: In the notation (B/A), the "/" symbol is not a division sign. (B/A) means "the number of ways B can occur given that A has occurred)

OR:

1.) p(A or B)=p(A) + p(B) Mutually exclusive: no overlap
2.) p(A or B)=p(A) + p(B) - p(A and B) Not mutually exclusive: overlap exists

Example 1:
a.) Sally has a crush on Billy and Fred. Each morning on her way into homeroom she passes the two boys: Billy is always at the lockers and Fred is always around the corner by the drinking fountain. The two boys can't see each other, so the probability of each one waving to Sally is not impacted by the other's actions. If there is a 50% chance that Billy will wave to Sally (he's sometimes in a sour mood) and there is an 80% chance that Fred will wave to Sally, what is the probability that tomorrow morning, both boys will wave to Sally?

Solution: Since the boys' actions are not impacted by one another, this is an independent event, and thus we have to follow the #1 equation under AND. All we do is plug in: p(B) x p(F)=.5 x .8=.4 >>> 40%

b.) Fred began to realize that the object of his affection was toying with his emotions, so now he is spying on Sally. If Billy waves to Sally at the lockers, there is only a 20% chance that Fred will then wave to her at the drinking fountain. If Fred catches Billy in the act, what is the probability that he will take pity on silly Sally and wave to her?

Solution: Because p(F) is affected by p(B), we now have to use equation #2. Since the question already gave us the probability of F given B, we plug in: p(B) x p(F)=.5 x .2=.1 >>> 10%

Complementary Events
p(A) + p(not A)=1
~ Note: This equation is true because in a given situation, the possibility of "A" and the possibility of "not A" are the only options available, and thus when added, they equal 1.
Example 2:
Probability that Billy waves: 50%
Probability that Fred waves: 80%
a.) What is the probability that Fred does not wave?
Solution: Given the equation above, we switch it around and plug in our value of 80%: 1-.8=.2 >>> 20%
b.) What is the probability that only one of the boys waves?
Solution: In this situation, we should first calculate the two individual possibilities (Billy waves and Fred does not wave or Billy does not wave and Fred waves).
Billy waves and Fred does not wave: .5 x .2=.1 >>> 10%
Billy does not wave and Fred waves: .5 x .8=.4 >>> 40%
Now we add the two probabilities together: 10% + 40%=50%
Lucy-you're next!!
This is a poem that I read last year in Ms. Yelverton's class. It's in a huge anthology of modern poetry, which I took off my shelf this weekend so that my sister could borrow it for her poetry project, and I "wasted" about 30 minutes of valuable SAT time going back through the poem. This is one of my favorites.
Having a Coke With You
is even more fun than going to San Sebastian, Irun, Hendaye,
Biarritz, Bayonne
or being sick to my stomach on the Travesera de Gracia in
Barcelona
partly because in your orange shirt you look like a better
happier St. Sebastian
partly because of my love for you, partly because of your love
for yoghurt
partly because of the fluorescent orange tulips around the
birches
partly because of the secrecy our smiles take on before people
and statuary
it is hard to believe when I'm with you that there can be
anything as still
as solemn as unpleasantly definitive as statuary when right in
front of it
in the warm New York 4 o'clock light we are drifting back and
forth
between each other like a tree breathing through its
spectacles
and the portrait show seems to have no faces in it at all, just
paint
you suddenly wonder why in the world anyone ever did them
I look
at you and I would rather look at you than all the portraits in the
world
except possibly for the Polish Rider occasionally and anyway it's
in the Frick
which thank heavens you haven't gone to yet so we can go
together the first time
and the fact that you move so beautifully more or less takes
care of Futurism
just as at home I never think of the Nude Descending a Staircase or
at a rehearsal a single drawing of Leonardo or Michelangelo
that used to wow me
and what good does all the research of the Impressionists do
them
when they never got the right person to stand near the tree
when the sun sank
or for that matter Marino Marini when he didn't pick the rider
as carefully
as the horse
it seems they were all cheated of some
marvellous experience
which is not going to go wasted on me which is why I'm
telling you about it
-Frank O'Hara