Finding the Proability of Multiple Events

Finding the probability of multiple events. Requires one to have a knowledge of simple probabilities. Below you will find a video lesson that defines independent events, dependent events, mutually exclusive events and inclusive events. Along with defining these types of events, how to find the probability of these types of events.

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Proof in Algebra – Geometry Meets Algebra

In a previous blog post about proofs in algebra, I wrote about how to a geometric style proof to solving an algebraic equation. The video embedded into this post models how to justify the solving of an equation with a two-column proof and the justifying of two problems involving the segment addition postulate and the angle addition postulate. Introducing proofs with algebra makes sense. You build upon your knowledge of solving equations to learn the new skill of creating proofs.

To be able to write proofs effectively, you must know definitions, postulate, properties and theorems. As always, I hope this information in the video and blog is helpful. Feel free to comment or ask a question.

Combinations and Permuations – Counting Methods

I wrote a couple of posts about combinations and permutations. These posts included definitions and examples of working with both permutations and combinations, but I know reading them is not always the best way for students to learn, so here is a video lesson of the two counting methods: combinations and permutations.

Geometry in Algebra – Using Proof in Algebra

You should be a student of geometry or algebra if you have found this post about using proof in algebra. If you are not a student of geometry or algebra, I encourage you to stick around and continue reading, you might just learn something. To effectively use proof in algebra and geometry, it is important to be familiar the properties of equality.

Properties of Equality

Properties of Equality

You should not forget about the distributive property of equality.

The Distributive Property

The Distributive Property

If you have taken algebra 1 or have experience solving equations, you already know these properties, but you are probably not used to naming them when solving an equation. This discussion about using proof in algebra is important, because many of the things you will asked to prove in geometry will involve solving equations, which is rooted in algebra. Here is an example of solving an equation in the form of a two-column proof.

Example – Justify each step in solving the equation (3x+5)/2 = 7.

proof-in-algebra-pilarski

Two Coloum Proof of Solving an Equation

Calling the problem above a proof is just a fancy way to say solve the equation. Instead of “showing your work” when solving the equation, you justify each new line, you use the two-column proof set up. Each step can be justified with a property of equality, which is seen in the reasons column of the two-column proof. Here is an example of solving a absolute value equation by “showing your work”. That is to say you are balancing the equation by performing the same operation to each side of the equation until the variable is isolated. Thus, the solution of the equation.

As has been seen with the segment addition postulate and the angle addition postulate, the use of algebra is integral. Without algebra, studying geometry would be more difficult. Most problems in geometry involve the use of algebra and learning how to “prove an algebra problem” will be a necessary part of making a proof in geometry, so, well it is a good idea to learn how to prove an algebra problem.

Segment Addition Postulate

Here is a video about the Segment Addition Postulate. I made the video in the fall of 2009. I was still really new to making math videos at the time. I have been making more videos this year. Well, I have been blogging about math more this year. Once I get to this topic again, I will be sure to create some images and write a good post.

Combinations

Related to the permutation is the combination. Both permutations and combinations are counting methods, but there is a distinct difference. Remember, when using a permutation the order is important. That is to say if you are to order a pepperoni, onion and mushroom pizza or you ordered a mushroom, pepperoni and onion pizza. Most people would say that is the same pizza, but if one thinks as the previously mentioned pizzas are different, then the person would have to use a permutation to count the situation.

My home made pizza. I usually make two!

Speaking of pizza, here is my home made pizza. I usually make two!

Counting with combinations means that order is not important and just rearranging the items does not count as a different combination. As can be seen in the formula. There is an extra factor in the denominator. I say extra factor in comparison to the permutation formula.

Combination Formula

Combination Formula

The additional factor of r! in the denominator divides out the counting different arrangements of the same objects as separate. Thus combinations usually give a smaller number of arrangements. Since I like pizza so much, here is an example that involves pizza.

Example – Combinations

Nate and wife like to go out to have pizza every Friday night. They to their favorite pizza joint Marcello’s located downtown. Marcello’s offers fifteen different toppings and has a special deal on 5 topping pizzas. The special is a large thick crust pizza loaded with 5 toppings. Regular crust could not hold all the pizza toppings. All this pizza goodness for $12.99. Anyway, Nate and Janet started talking about how many different pizzas there were to choose from at Marcello’s. Help them figure it out.

Since the order of the toppings on the pizza does not matter or make it a different pizza, a combination should be used. There are 15 toppings to choose from and they are allow to have 5. So, one must take 15 toppings taken 5 at a time.

15 pizza toppings taken 5 at a time.

15 pizza toppings taken 5 at a time.

Below is my best attempt at a quick photo shop job to work out the above combination with all of the work shown.

15combo5work-pilarskiIf this has helped you in anyway, leave a comment. If you still have a question you can comment your questions and I will be happy to answer you.

Permuations

Any mathematics textbook you look you will find a definition of a permutation.The thing is, no matter where you look, they will all be the same. Permutations and Combinations are a ways to count a sample space.  There is a major difference between the two.

Permutations are derived from the fundamental counting principal, but you will choose all of the items in the set. For example if you have 5 books to put on a shelf. For the first book to shelf you 5 to choose from and you shelf one of them. Now you have 4 to choose from and you shelf one of them. Now you have 3 to choose from and you shelf one of them. Now there are 2 and you shelf one of them and now you only have 1 to shelf. Well, the fundamental counting principal would give:

Example: 5! or 5 Factorial

Example: 5! or 5 Factorial

This type of multiplication is referred to as a factorial. The above example would be read as “five factorial” and written as 5!. But life is full of choices and many teams have more players than starters. Knowing how many ways you can arrange a team can be helpful with coaching decisions. Do you know how many ways a coach can arrange 5 starters on a basketball team when there are 11 players to choose from? It is not a guessing game either. If you don’t know that is okay. Think about the only two numbers mentioned in the problem. 11 players to choose from but only 5 starting spots. Below is the formal definition of the number of permutations.

The number of permutations of n items of a set arranged r items at a time is

Definition of Permuations

Definition of Permutations

The great thing about living in today’s information age is the technology. It is every where. The nice thing about technology is the pain staking task of finding permutations can now be done by calculator. Solving these types of problems could not be easier. Suppose you there were 14 wrestlers competing in a tournament and the top 4 places were awarded a trophy and prize money. In this situation, the order in which the wrestlers is important to all parties involve. The organizers need to know who to pay and the wrestlers care where they place because the higher the place the higher the prize money. You would count the different ways the 14 wrestlers could finish in the top 4 using a permutation. P(14, 4) = 24, 024 ways the wrestlers could finish.

There are many other situations you could create or may even run into, where counting the sample space is necessary or fun! Soon to come, a post about combinations and a video on both permutations and combinations.

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