Bittersweet Week

So I got the call this morning, and Better Candidate outshone me again because they had more experience in a classroom setting. Tell me universe - how am I supposed to get more experience when I can't get into the classroom?

I've also been dealing with the most painful ear infection I can recall. The doctor gave me steroidal ear drops but now my ear feels like it is full of water and I would punch kittens if it would make the pain and stuffy-ear-feelings stop.

On the bright side: I got a perfect score on my first math test of the semester.

I have never received a perfect score on a math test in my life, so I'm a bit excited. Part of it I'm sure has to do with reviewing the material several times in order to teach it to other students and explain it in the blog. I also have get to start making a game for an elementary level class involving statistics. We'll see how that one goes. Now if only I can do as well on my art history exam today as I did on my math test - fingers crossed!

Odds

So I'm on a math roll today, finally got my math homework done and realized that I am one math post short for the math blog - oops! So I'll be covering the last and shortest part of the probability section: Odds!

Now, odds and probability are different, and it is important to know what that difference is.

Image from Math Magic website

The numerator for both is the amount of times you obtain the desired outcome, but the difference lies in the numerator. While probability has a denominator that lists all the outcomes, when calculating odds, you have the undesirable outcomes as your denominator.
To find out what the undesirable number is, simply find the total number of outcomes and subtract from that the number of desirable outcomes. This leaves you with the undesirable outcomes, your odds denominator.

Math Magic has a great page explaining probability and how to change from probability to odds. See it here: Math Magic - Probability

Probability Game

This probability game can be adjusted for any grade level, and makes for a great online practicing resource!

Probability Game

Probability (Part II)

I began writing this post a week ago and forgot to publish it! I could kick myself :P

Let me start with this: Tree diagrams are not my friend. They confuse me to no end and I couldn't tell you why, they just do. So when we began our probability section, and tree diagrams were introduced as the way to record information I just about lost it. It took me three lectures for it to click, and even now I couldn't explain it to with total confidence, but here we are. Geronimo!

Multistage Experiments with Tree Diagrams

• One-stage experiment - experiments that are over in one step.
• Two-stage experiment - Ex.: a ball is drawn from a box, and recorded in the same way as in the one stage experiment; then the ball is replaced, and a second ball is drawn and its' color recorded. 

Now, tree diagrams are used to record the results of these experiments using visual representation in lieu of boring old data tables. When recording and performing there are two types: with and without replacement. They are exactly as they sound - in one you perform the experiment and replace what was removed, and in the other you don't.

"Pom Pom" experiment showing probabilities
both with and without replacement.

Above is one such experiment, using puff/pom pom balls. In a box we have 3 balls, two white and one black. The probabilities are calculated based on the ratios: 2/3 of the balls are white, 1/3 are black. Depending on whether or not we replace the balls after pulling them from the box changes the probability of drawing that color next. We determine that probability by multiplying the ratios.

Another example is shown here:

The first example shown in the image illustrates another probability concept using the word PROBABILITY itself. It demonstrates the probability of spelling the word BABY, with the letters drawn in the correct order, both with replacement of the letters and without replacement. Again, we multiply to find the probability.

The second example uses the same idea, only using the word RANDOM to spell DAN. Again this shows both with and without replacement.

So we've been talking about probability, but what is it?

So probability is fractions (kind of), which may explain why I've had so much trouble - fractions are not my friends.

Bill Nye Probability Episode

Bill knows what's up.
He may be the "Science Guy", but he is pretty handy when it comes to math.

Probability (Part I)

So I'm about a week (or so) behind on updates, but I'd like to think I spent the time putting together an amazing wealth of knowledge in preparation for my first math-centric post. Though I would not necessarily deny any accusations of procrastination to avoid my arch nemesis for as long as possible.

Used with permission from MathFunny.com

Math and I have never been the best of friends, especially during times when probabilities and/or statistics are involved. But the information we've been going over in class the last couple of weeks has helped to make me (albeit, a very small amount) more confident.

How Probabilities are Determined

We begin with the basics: vocabulary terms. Often the definition of words change depending upon what field you are using them in; things that have one meaning in layman's terms will have a completely different meaning in science, math, and especially law. Math is like another language entirely, and studying it is essentially learning the definitions of each word or equation. Each formula, equation, etc., has its' own definition which tells you what it means and how to properly use it.

• Experiment - An activity in which results can be observed and recorded.
• Outcome - Each of the possible results of an experiment. (Ex.: With coins the outcome is heads or tails)
• Sample Space - A set of all possible outcomes for an experiment. Results such as S = (H,T) can be modeled by a tree diagram, which we will look at later.
• Event - Any subset of a sample space, such as the event of dice; an example of this would be 
  S = (2, 4, 6) where the sample space for S is rolling a standard die with even numbers as the outcome.

When determining probability, there is Experimental (or empirical), and Theoretical. the difference between the two is that experimental is what you actually observe while theoretical is what would happen under perfect and ideal conditions. Additionally, we believe in and try to abide by Bernoulli's Theorem which states that if an experiment is repeated enough times that the empirical probability will become closer to the theoretical probability. Sort of like the whole monkeys and typewriters idea.

We encounter from time to time events that are impossible or certain. These are represented in an equation as  0 and 1, respectively. If something is equally as likely to happen as another option, we often describe that as being a 50/50 chance or probability, however we would write that as a 1/2 probability. Events can also be mutually exclusive or complementary. With mutually exclusive events if one thing happens the other cannot, OR they are mutually exclusive if the events have no elements in common. Complementary events are along the lines of: Chance of rain = 25%, so the chance of no rain is 100% - 25% = 75% or 3/4 probability of no rain. 

On a somewhat related note, the probability of many home
owners in the Phoenix Metro Area getting flood insurance after
this weeks crazy, valley-wide flooding is very high. 
Image courtesy of the Associated Press & BBC World News

The following websites do a much better job at explaining introductory probability than I do:

Math Is Fun

Khan Academy

Math Goodies