Week 12: The Probability of an Event

Probability is somethign that has been looked at multiple times throughout ones childhood. There’s the probability of getting 5 cards out of 52. There’s the probability of finding 1/2 of a goldfish than a whole goldfish. There is multiple types of probability.

Though these are some words needed to know probability

Outcome means experiments result

sample space means a set of all outcomes (s)

event means the subset of sample space (E) or what you want to happen.

then there is a probability of an event, which is

P(E)=n(E)/n(s) or # of ways E can occur/ # of ways of all outcomes

The easy way of probability is trying to find the probability of dices on certain amounts like

ex: on a single roll of a fair dice, find the probability of:

a) rolling of a 4: 1/6

b) rolling an odd #: 1/2

c) a # that is not a 4: 5/6

But there are harder ones too using nPr or nCr. 

For example. 

A bag contains 5 white marbles & 3 red marbles. A person draws 3 marblse. What is the probability all marbles are white?

n(white)/ n(draw 3)

C(5, 3)=10

C(8, 3)=56

so its equal to 10/56 or 5/28

Then there is a complement of an event which is a set of outcomes that do not belong to an event (E^1)

P(E^1)=1-P(E)

One example is:

If 5 cards are drawn from a deck of 52, without replacement. Find the probability of at least one ace?

P(no aces)=

C(48, 5)= 1712304

C(52, 5)= 2598960

1- 1712304/259860

=0.341158

Then there is mutually exclusive which is 2 events with no outcomes in common so what you do is look for what they do have in common. 

This is what you use to solve them:

P(E1 U E2)=P(E1)+P(E2)-P(E1upside down Union E2)

the U is union and the upside down union is called interaction

Above are examples on how you solve. 

That is how you do probability.

SEE YOU NEXT WEEK!!!

Week 11: The Tower of Hanoi

The Tower of Hanoi is a mathematical puzzle that consists of three rods and a number of disks of many different sizes. You have all the disks to the last rod on the right without putting a bigger disk on top of a smaller disk.

These are the rules:

1. Only one disk can be moved at a time.

2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another disk.

3. No disk may be placed on top of a smaller disk.

This puzzle was created by the French Mathematician Edouard Lucas in 1883. However, there is a story in Indian temple in Kashi Vishawanath that there is a large room with three worn posts surrounded by 64 golden disks. There is a prophecy where when the puzzle is solved and the last move of the puzzle the piece is complete, the world will end. This sounds fake and somewhat frightening, however, it would take some time till then.

It would take 585 billion years to actually finish this puzzle of moving all 64 pieces. It would take 18,446,744,073,709,551,615 moves to actually finish this at a minimum moves possible.

The puzzle is 2^n -1 where n is the number of disks. That is how you solve it.

In class today, we used mathematical induction to solve this and using an app to understand mathematical induction through the tower of hanoi. That is what we did this week.

Here is a picture on how we solved it.

SEE YOU NEXT WEEK!!!

PI DAY!!!!

HAPPY PI DAY!!!!

IT’S SUPER PI DAY!

3.14.15

MARCH 14, 2015

3.14159265

MARCH 14, 2015 AT 9:26 AM!

HAPPY PI DAY!!!

Week 10: Sequences and Series

This week we looked at Sequences and Series.
A sequence would look like
a1,a2, a3...an,an+1

To find out how a sequence runs you plug it into the equation that it is given in to n and find a1, a2, etc.

Another type that is looked at is Recursion. Recursion is the use of the term before it so for example a4 would be a3.

So the Arithmetic Sequence which is adding and subtracting has two equations the general and the recursion.
The General equation is: an=a1+(n-1)d
The Recursion equation is a(n+1)=an +d
the d stands for common difference and you can find that by a(n+1)-an
Below are two examples of problems that there are.

The first one is you must figure out the difference by plugging in a1, a2 etc.

The second one you have to find the first term of the sequence by plugging in information given.

Then there is the geometric sequence which is multiplying.

The General equation is an=r^(n-1)

The Recursive equation is a(n+1)=anr

r stands for the common ration and to figure out the common ration you use a(n+1)/an

Below are two more examples on how to plug in and solve for the rate and finding the first term in the sequence.

We then looked at Compound Interest. The equation for compound Interest is An=P(1+r)^n

P stands for Principle or how much you start with

R stands for Interest Rate which needs to be in a decimal

n stands for the number of years

An=the amount you get back after n years

To solve you simply plug in and solve for the necessary number needed.

Then we looked at Series. 

A Key piece of series is Summation Notation which uses ∑. 

In the above picture of the symbol. It means sum and to solve you plug in the amount for k into the equation how many times the n says.

Below are two examples of how to solve.

Then you get into problems on how to write the summation notation out of a series of numbers.

In the above equation, you first find out whether it is arithmetic or geometric and then find out the rate or distance. After that you plug it into the sequence equation  and then you look for the n. That is how you solve the equation.

There also is the Arithmetic Series

Sn=n(a1+an)/2

n is how many terms and Sn is the sum

There also is the Geometric Series equation.

Sn=a1(a-r^n)/1-r

Lastly here are some properties that are very important in summation notation.

That is what we learned this week.

SEE YOU NEXT WEEK!!! 

*Don’t forget Pi Day*

Week 9: Graphing Systems of Inequalities

This week we covered Graphing Systems of Inequalities in the chapter Systems of Equations and Inequalities. 

This is simply review from Algebra from graphing the inequality and shading. 

First you will:

1. Graph each equations

     a line: y=mx+b

     a parabola: y=(x-h)^2 +k

     a circle: x^2 + y^2 = r^2

2. Pick a test point not on the line

3. Shade the plain containing the test point. If the test point satsifies the equation shade the other plane ift id does not

   ≥ or ≤ solid line

   > or < dotted line

This is how you will solve such a problem.

Below is one example of this problem.

In this problem. You can see the equations being graphed and then finding which side to shade. the place where both lines have shaded is the answer for the equation. 

That is how you graph systems of inequalities

See You Next Week!!!