Warren Buffett and the owner of the Cleveland Cavaliers are offering one billion dollars to anyone who gets a March Madness (college basketball) bracket perfectly right. This might seem like an outrageously high number, but interestingly enough, they may not even have to pay it! Let's calculate the odds.
1. There are 68 teams that make it into the tournament, and since each team loses once except the winner, there are total games.
2. Let's assume that you have about a 50% chance of guessing each game correctly. [This might even be an understatement, because some really good teams will play really bad teams (especially early on) and will have a more than 50% chance of winning.]
3. After each successful guess, your next guess will also be a 50% chance, so your chance of winning the whole thing is 0.5 x 0.5 x 0.5 x 0.5 ... until you get 67 0.5's, so (0.5)^67.
4. (0.5)^67 = 6.78 x 10^-21, or 0.00000000000000000000678. That's about a one in six sextillion!
Now, most experts think that it is possible to guess the outcome of some games with such certainty that the average percentage of guessing a game right could be closer to about 75%. With this calculation, your odds are still about one in four billion. Buffett is only allowing 10 million entrants, so in all likelihood he won't even have to give the check to anyone!
Works Cited:
http://parade.condenast.com/255536/erinhill/hes-gone-mad-warren-buffett-offers-1-billion-for-perfect-march-madness-bracket/
Thursday, February 27, 2014
Binomial Expansion
Binomial expansion is basically the opposite of factoring. It takes a basic, fully factored combination of variables and numbers and expands it from a basic form (ax +by)^n. The following picture shows a binomial expansion of the equation (2x - 3y)^4:
To find out the coefficients of each term, use Pascal's Triangle (shown on top). Then, take the first term (ax) and with the first coefficient, set it to the 4th power, then the 3rd power, etc. down to the 0th power. Do the opposite with the 2nd term (by), starting at 0th and then going up to the 4th power. Finally, multiply the terms to simplify. Final answer is in green.
Wednesday, February 26, 2014
Well-Ordering Principle
According to Wikipedia, the well-ordering principle states that every set of positive integers must contain a lowest or least element. This is a very key element in validating mathematical induction as a way to prove formulas.
IMPORTANT: The goal of mathematical induction is to prove that for any number n, a given equation would find the sum of a given sequence.
The first step in mathematical induction is to prove the equation works if n=1. This is possible because of the well-ordering principle, because it proves that n=1 is possible by saying that every positive-integer set has a first number. The next step is to test for n+1, as since we know the equation applies for n=1, this would prove it applies to n=2. But in that case, since n can equal 2, it proves it for n=3, then n=4, and so on and so forth, proving the equation to be true regardless of the value of n.
Works Cited:
http://en.wikipedia.org/wiki/Well-ordering_principle
IMPORTANT: The goal of mathematical induction is to prove that for any number n, a given equation would find the sum of a given sequence.
The first step in mathematical induction is to prove the equation works if n=1. This is possible because of the well-ordering principle, because it proves that n=1 is possible by saying that every positive-integer set has a first number. The next step is to test for n+1, as since we know the equation applies for n=1, this would prove it applies to n=2. But in that case, since n can equal 2, it proves it for n=3, then n=4, and so on and so forth, proving the equation to be true regardless of the value of n.
Works Cited:
http://en.wikipedia.org/wiki/Well-ordering_principle
Thursday, February 20, 2014
Comparison of NBA MVP Candidates
Four of the MVP favorites in the NBA this season are Paul George, LeBron James, Kevin Durant, and Blake Griffin. One simple way to test a player's offensive ability is to measure their points per game:
1. Kevin Durant: 31.5 PPG
2. LeBron James: 26.8 PPG
3. Blake Griffin: 24.4 PPG
4. Paul George: 22.5 PPG
But this statistic will not factor in how often a player shoots the ball. If a player shoots 30 shots per game, it is a lot easier to score 30 points. In this case, a player would only have to make 50% of their baskets to score 30 points just off of two-point field goals, not even taking into account 3-pointers or free throws.
1. LeBron James: 57.4% FG
2. Blake Griffin: 54.0% FG
3. Kevin Durant: 51.0% FG
4. Paul George: 43.9% FG
Now, these statistics clearly only factor in offense. It is difficult to factor in defense with statistics, as the only concrete statistics in this category are blocks and steals. In my opinion, the best stat would be a measure of the field goal percentage of the opposing player whom a player is covering. Unfortunately, this statistic is unavailable on ESPN.com, but it is generally agreed that this is the order of these four players from best defender to worst defender.
1. Paul George
2. LeBron James
3. Kevin Durant
4. Blake Griffin
Since all these players excel in all statistics, it is difficult to use statistics to separate one from another. George is the best defender, but the worst scorer of the four. Most analysts agree that LeBron James and Kevin Durant are the only players with a shot at winning MVP, but even though Durant holds a clear edge in points per game, James' edge in field goal percentage is far more of a deciding factor, because it shows that he knows what is best for his team in every situation and only shoots high-percentage shots. James also has an advantage when it comes to intangibles, because he has proven that he has the talent and leadership to win in the playoffs. Therefore, despite Durant's astronomical scoring numbers that cannot be overlooked, I think James should win his 5th MVP in 6 years.
1. Kevin Durant: 31.5 PPG
2. LeBron James: 26.8 PPG
3. Blake Griffin: 24.4 PPG
4. Paul George: 22.5 PPG
But this statistic will not factor in how often a player shoots the ball. If a player shoots 30 shots per game, it is a lot easier to score 30 points. In this case, a player would only have to make 50% of their baskets to score 30 points just off of two-point field goals, not even taking into account 3-pointers or free throws.
1. LeBron James: 57.4% FG
2. Blake Griffin: 54.0% FG
3. Kevin Durant: 51.0% FG
4. Paul George: 43.9% FG
Now, these statistics clearly only factor in offense. It is difficult to factor in defense with statistics, as the only concrete statistics in this category are blocks and steals. In my opinion, the best stat would be a measure of the field goal percentage of the opposing player whom a player is covering. Unfortunately, this statistic is unavailable on ESPN.com, but it is generally agreed that this is the order of these four players from best defender to worst defender.
1. Paul George
2. LeBron James
3. Kevin Durant
4. Blake Griffin
Since all these players excel in all statistics, it is difficult to use statistics to separate one from another. George is the best defender, but the worst scorer of the four. Most analysts agree that LeBron James and Kevin Durant are the only players with a shot at winning MVP, but even though Durant holds a clear edge in points per game, James' edge in field goal percentage is far more of a deciding factor, because it shows that he knows what is best for his team in every situation and only shoots high-percentage shots. James also has an advantage when it comes to intangibles, because he has proven that he has the talent and leadership to win in the playoffs. Therefore, despite Durant's astronomical scoring numbers that cannot be overlooked, I think James should win his 5th MVP in 6 years.
9.2 Arithmetic Sequences
This chapter focused on arithmetic sequences and how to find the sum of a sequence. This problem combines finding the arithmetic equation for a series and finding the sum of said series. In this problem, we are trying to find the sum of a set of 25 numbers. First, we must find the common difference to complete the equation for the arithmetic sequence equation shown below. Then, we can find a25 using that equation. Finally, we plug in a1, a25, and n (which is 25) into the sum of an arithmetic sequence equation shown at the bottom, and we get 1850 as the answer!
Wednesday, February 19, 2014
9.1 Sequences and Summation Notation
A few of the things we learned to do in this lesson are:
1. Find terms of a sequence.
2. Write equations of sequences.
3. Evaluate factorials.
1. Sequences are often represented as an equation like an = 2n - 1. To find a specific term, like the third term (a3), plug the number of the term into the equation as n. 2(3) - 1 = 5.
2. Finding the equation of a sequence can be tricky, but the best method is to just list the sequence with the number of each term under it., like so: sequence: 1 3 5 7
n: 1 2 3 4
The pattern can be determined from there.
3. A factorial (!) is a product of consecutive natural numbers, leading up to the number by which the product is represented.
For example, 5! = 5x4x3x2x1.
A lot of times equations that involve factorials can appear overwhelming at first, but simplification is possible if there is division involved:
Example: 100!/99! -----> The denominator cancels out the first 99 factors of the product 100!, so the only number left is 100. Therefore, the answer is 100.
Most of these problems involve a lot of logical thinking rather than plug-and-chug equations. This is good for some students who understand that logic, but it can be a struggle for other students.
1. Find terms of a sequence.
2. Write equations of sequences.
3. Evaluate factorials.
1. Sequences are often represented as an equation like an = 2n - 1. To find a specific term, like the third term (a3), plug the number of the term into the equation as n. 2(3) - 1 = 5.
2. Finding the equation of a sequence can be tricky, but the best method is to just list the sequence with the number of each term under it., like so: sequence: 1 3 5 7
n: 1 2 3 4
The pattern can be determined from there.
3. A factorial (!) is a product of consecutive natural numbers, leading up to the number by which the product is represented.
For example, 5! = 5x4x3x2x1.
A lot of times equations that involve factorials can appear overwhelming at first, but simplification is possible if there is division involved:
Example: 100!/99! -----> The denominator cancels out the first 99 factors of the product 100!, so the only number left is 100. Therefore, the answer is 100.
Most of these problems involve a lot of logical thinking rather than plug-and-chug equations. This is good for some students who understand that logic, but it can be a struggle for other students.
Thursday, February 13, 2014
Thursday, February 6, 2014
Cryptography
On Wednesday, we learned about a form of cryptography, or code language, that uses matrices. In the first step of this worksheet, we had to use an encryption matrix and code Miss V gave us to find out the meaning of the code. We had to use the inverse of the encryption matrix, which is its decryption matrix, to decode the message into new numbers (3 numbers at a time to multiply by the 3X3 decryption matrix). Finally, we had to transfer those new numbers into letters (A=1, B=2, etc.), and the code spelled out the words "MEET ME MONDAY". I then created an encryption matrix of my own in step 2 and made a code. Can you figure out what the code means?
Wednesday, February 5, 2014
Recognizing Type of Solutions to Systems of Equations
Solving systems of equations using matrices is very helpful, as there are several tricks to be utilized to simplify the system quickly. The most common use of the Elementary Row Operations are to simplify equations into row echelon form.
Row Echelon Form Requirements:
1. All rows consisting entirely of zeros occur at the bottom of the matrix
2. For each row that does not consist entirely of zeros, the first nonzero entry is a 1.
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.
(b)
Sources:
http://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx
Row Echelon Form Requirements:
1. All rows consisting entirely of zeros occur at the bottom of the matrix
2. For each row that does not consist entirely of zeros, the first nonzero entry is a 1.
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.
From row echelon form, we can find the type of solutions the system will have. In the following example, we can tell that the system will have infinitely many solutions from the row echelon form.
(b)
Since the bottom row would transfer into the equation 0 = 0, the other equation x + (5/2)y = -1/2 is the only relevant equation. Therefore, any number x will result in a number for y, so there are infinitely many solutions.
Here's another example:
If you switch the final two equations at the end of this example, it is in row echelon form. That final line would transfer into the equation 0+0+0=12. Obviously, this equation is incorrect, so the system as a whole is inconsistent and therefore has no solutions.
Sources:
http://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx
Monday, February 3, 2014
Inverse Matrices
In both of these examples, I'm attempting to find the inverse of the original given matrix. The idea is that a matrix times it's inverse will equal the identity matrix. The identity matrix is a matrix with all zeroes, but a main diagonal of all 1s. To complete these problems, write the matrix along with an identity matrix in the same box. Then, simplify until the only nonzero numbers are ones on the main diagonal, just like you would do with a normal simplifying matrices problem. The first of these examples didn't work because there was a row of zeroes, so it couldn't be out in the proper form of only ones on the main diagonal. This means that the system is singular, not inversible, and does not have an inverse.
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