9.8 and 9.9 - Exploring Data

These sections expand on the main ideas of mean, median, and mode that we've known for a long time. But now, Variance and Standard Deviation are added. The equations for these are given in the chapter, but most of the problems involve too many numbers to do in a reasonable amount of time, so it's a good decision to just use standard deviation and variance calculators online. The following example finds the mean, median, and mode of the set 3, 7, 14, 15, 21, 21.

Mode: 21

Median: 14.5

Mean: 13.5

9.7 Probability

Probability problems can seem really complicated, but are often a lot simpler than they appear. At the base, every problem is asking you the ratio of favorable outcomes to total circumstances 

(# of successes/# of possibilities). For example, if you want to get a coin flip to be heads, there is one successful outcome, but two total outcomes (heads and tails), so the probability of getting heads is 1/2. There are twelve face cards in a deck of 52 cards, so the probability of drawing a face card is 12/52 (3/13). Let's try an example:

14.) In a 52-card deck, one card is drawn. What are the chances the number is six or lower?

• Ace, 2, 3, 4, 5, and 6 are six or lower, and there are four of each of those, so 6X4=24.
• There are 52 total cards, so the probability is 24/52 (Simplified: 6/13).

March Madness Odds

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/

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.

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

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.

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!