The Flip of a Coin

What is the probability of flipping a coin and it landing on heads? 1/2
What is the probability of flipping a coin and it landing on tails? 1/2 also. 

Well Bernoulli's Law of Large Numbers says that the more you perform an experiment many many times, the closer the experimental probability result will come to match the theoretical probability. 

Our professor taught us how to simulate a coin toss on our TI-83 calculators so we could simulate the event happening opposed to having 25 people flipping a coin 40 times each. Here are some tutorials on how to perform that function on your calculator with a TI-83 or a TI-84. 

For my 50 trials, I came up with 24/50 or 49% heads and 26/50 or 51% percent tails which is very close to the theoretical probability 25/50 or 1/2 or 50%.

Together as a class we added all of our totals together with a grand total of 700. According to Bernoulli's Law, it should approach much closer to the theoretical probability of 1/2 or 50% for both heads and tails. 

BUT I got pretty close the first time, being one percent off on both heads and tails. As a class we got 327 heads (47%) and 373 tails and (53%). Still very close to being half and half. 

Simple, but a great easy way to look at theoretical and experimental probability!

Make Like A Tree and Leaf Those Worries Behind

Tree diagrams definitely provide the connection that I had always been missing to understand when to multiple events or add them together. 

Making a tree graph is a great way to to physically see how many different outcomes of an event can occur in an organized fashion. Another advantage to tree graphs, is an easy way to find points of misunderstandings. 

Let's use rock, paper, scissors to make a tree diagram for calculating theoretical probability! 

So starting off making the first branches. 

You know you are going to begin with three branches because you have three choices, rock, paper, or scissors. You can choose one out of those three so the probability for each  one is 1/3. Continuing, to see what the probabilities are if you pick any one of those, we have to add in the chances of what your opponent will do! 

So once you pick paper, your partner also has 3 choices of what they can do, rock, paper, or scissors. You would draw 3 branches off of paper and label them P,R,S. Since your partner also had three choices, you label each one with 1/3. 

You repeat these steps for each one until your tree diagram is complete. 

As you are moving across the tree diagram, you multiply each event times itself. 

Paper (1/3) x Paper (1/3) = A tie (1/9)

Paper (1/3) x Scissors (1/3) = Scissors wins (1/9)

Paper (1/3) x Rock (1/3) = Rock wins (1/9)

Scissors (1/3) x Paper = Scissors wins (1/9)

Scissors (1/3) x Scissors = A tie (1/9)

Scissors (1/3) x Rock = Rock wins (1/9)

Rock (1/3) x Paper = Paper wins (1/9)

Rock (1/3) x Scissors = Rock wins (1/9)

Rock (1/3) x Rock = A tie (1/9)

You notice that if you add off these events up (1/9) they come out to (9/9) which equals one. 

When I see branches on a tree, I can see that I should multiply the events together because they are multi-step, they are coming one of after the other, otherwise you probably wouldn't need to tree for it, eh? Well, after you determine a multitude of probabilities, you tend to see that all of your events have the same denominator. This was easier for me to understand that I need to add these numbers together because they are out of 1 essentially. Your probability will always be somewhere in between 0 and 1 because if it wasn't going to happen at all, whatsoever (0) then you wouldn't be willing to risk it. And if it was definitely going to happen (1) you would have done it already! So there you have it, it somewhere between zero and one. 

These end probabilities (1/9) will help you figure out theoretical probabilities. The probability it will end up as tie- (1/9)+(1/9)+(1/9)=(3/9)=*reduced*(1/3)

So you could guess your probability of a tie when playing rock paper scissors! :)
This man has a nice accent to listen to while educating you on how to execute the compilation of tree diagrams. But if your mom keeps asking you what you're doing on the computer and you need a break... you can be mildly entertained by this guy explaining probability. Now say "MOM! I'm calculating probability!" Technically you're still watching something educational and he's a weirdo, sounds like the internet to me!  If you watched it, I'll allow one cat video.. done? Okay back to my blog. 

Experimental vs. Theoretical

Understanding probability has always been a very difficult concept for me. Mostly because #1, I always dreaded fractions and #2 the way the word probably is overly used- mostly in lies. 

But as I have taken Algebra 3 times in high school, and 3 courses in college, I am getting pretty good at understanding math as something other than a jumble of number where you have to memorize formulas and theorems to even take a whack at what you're doing. I have begun to see the whole picture where all concepts connect and intertwine. 

It is important to have solid roots in math when examining experimental and theoretical probability and even the definitions of those two words. Theoretical probability is predicting what SHOULD happen based on the number of outcomes. Theoretical probability cannot calculate one big variable, humans! Experimental probability is relevant when one is conducting an experiment and calculate probabilities from the the results of that experiment. So it is what DID happen. Humans are quite often involved in experiments which tends to be the reason the two probabilities do not line up. 

Note to reader: If you do not know how to play rock, paper, scissors please click here!

We continued using our data from the Rock, Paper, Scissors for this activity. Here is the picture so you don't have to scroll or click around and get lost :)

FINDING THE PROBABILITY THAT I LOSE WITH ROCK AND MY PARTNER WINS WITH PAPER.

To start off, we calculated the probability that I will show rock. So I look at the bottom row and count up all of the times that I showed rock. I am Cortni if anyone forgot so I did rock 3 times + 6 times + 4 times for a total of 13 times out of 44 trials so that would give us a probability of 13/44 (0.3 in decimal form). 13 for the times it happened and 44 for the total number outcomes. 

Same thing for when my partner showed paper. 4 + 11 + 3 = 18. So the probability in this particular case that my partner would choose paper is 18/44. Or 9/22 (0.41 in decimal form) if we were to reduce. 

Now we can calculate what would seem like a fairly accurate theoretical probability since we are using the actual results from our game but let us see what happens. If we multiply the number of times I chose rock, 13/44 times the reduced number of times Hannah chose paper 9/22 we get a probability of 117/968 which cannot reduce but as a decimal is 0.12. Sometimes it is easier to convert fractions to decimals so it easier to compare. 

Last step! We look at the ACTUAL probability which was what DID happen. So for that we refer back to our matrix and look at the box that tells us that I lost with rock and my partner won with paper which was 3/44. Converted to a decimal is 0.06.

When we look at what our theoretical or predicted probability was that I lost with this combination it was 0.12 compared to what experimental or actual probability which was 0.06 you see that they were not close at all!

To explain this occurrence, I would have to brag that I noticed the pattern that was forming when Hannah would always play paper so I stopped playing rock and started playing scissors so I could win! And that is what I meant when theoretical probability doesn't count for human interference. And that is now how I understand the difference between theoretical vs. experimental probability!

Rock, Paper, Scissors

August 29, 2013

To continue with the theme of probability, we played Rock, Paper, Scissors today! 45 times. Actually I think my partner and I only got 44, but we had some human error interference. We marked some accounts differently but we aren't too competitive so it we just agreed to change our answers according to one person's paper. 

Note to self: Make sure to explain how to play rock, paper, scissors. You may have people in your class from other cultures or even your own culture who have never played rock, paper, scissors before. Therefore, it is important to review to rules of the game and how to play.

This activity was accompanied with a matrix to help keep track of results. 

The real point of this activity was for us to decide if rock, paper, scissors is a fair game. 

My results show that I won half of our trials and the other half were split  evenly between my partner's results and the occasions that we tied.

Honestly, I have never been good at this game. I hardly ever win. I do have to admit though, that when I noticed that most of my winnings were when I chose scissors, I starting choosing scissors to see if I would keep winning when my partner chose paper. Now I'm not sure it had to do with the fact that I was choosing scissors but more the idea that my opponent was partial to choosing paper. This is when I began to strategically choose my next move. I was not reading the "body" language of my opponent, only quickly assessing the results on the paper. 

In the end, I won :) I don't think this game was completely fair because of my sneaky observations.

Obviously these people know how to win every time without a matrix in front of them!

MATH CLUB!!

So today in Math Club, we played Farkle. I really do not like the name of that game but after it took me like 20 minutes to figure out how to play I realized why the slogan was "A Risk-Taking Game!" You had to know your odds. In a nutshell, you roll the dice for points. Generally, it is in your benefit to roll three or four of a kind so you had to calculate the probability of receiving the results beneficial to you to see if the risk is worth it, or to stick with what you have! 

That's what I was doing the whole time, calculating probability :) Made me feel like such a proud math club member!! 

Cut the Deck!

August 27, 2013

Note to self: When you get decks of cards, make sure you take out the jokers!

The lesson started with the students writing down everything they know about a deck of cards. Such as how many cards are in one deck, how many cards in each suit, what those suits are, how many face cards, etc. This was a good memory jogger to get your brain thinking in numbers for a minute. Then worksheets were handed out asking us to calculate probability without mathematical notation so it was easier to comprehend. 

I do like the idea of working with cards because it is yet another tangible item that can be utilized in the classroom. 

My favorite questions on this sheet were where it would ask for the probability of a queen, and then not a queen. It gets you used to the idea of the complement(make this word clickable) of one problem and where you can learn to take less steps to get to an answer and not make your brain so tired! 

P(Queen)= 4/52 = 1/13
P(Not a Queen)= 13/13 - 1/13 = 12/13

Next, we used small pom poms from a craft store to help us calculate probability with tree diagrams. I love visual strategies!

As much as I love mathematical formulas, I much better understand them when I understand WHY they work. Also, if I don't remember the formula quite correctly then understanding a concept can help me derive the equation myself and still arrive at the desired answer. 


Another important lesson I learned today was to understand the meaning of words! Ahhh, english in math, what a lovely sight!! I really do feel lucky that I have found a subtle if not extreme love for most general education subjects. 

AT LEAST means there is no maximum! 
AT MOST means there is no minimum! 

If the question is asking for the probability of at least drawing one white marble  from a bag then... at least would include all events where there are at least one marble in the outcome. So if there are 27846236 white marbles in a particular outcome, you count that one too!!

If the question is asking for at most three white marbles drawn from a bag then 1,2,3 and even ZERO white marble in the outcome counts! 

Go Fishing with Probability!

August 21, 2013

Today, I came to MAT 157: Mathematics for Elementary Teachers II prepared. Though, when walking into a classroom with a great teacher, you can only be so prepared because those teachers often surprise you. That is the type of teacher I aspire to be one day. 

We learned a new classroom management technique when placing students into groups which involved only a stack of playing cards. You take two-of-a-kind out of the deck to match the number of students present in the room so they may match themselves together. This method takes a little bit of preparation but I appreciate the exposure for when classroom procedures become monotonous. Note to self: Roxanne also warned us that older children will start switching cards, so you must be strategic when handing out the cards. Maybe the beginning of the class as the students walk in.

Our lesson was a slight overview of probability and a strategy to teach it involving goldfish. The snack, not the water creature. A ziploc bag with the colorful goldfish were handed out, all different totals. A general worksheet accompanied the activity that required students to predict probabilities of sample scenarios and then apply those predictions. 

Specifically my partner and I received a total of 47 goldfish. 15 yellow, 15 orange, 7 red, and 10 green. 

My partner and I got through the worksheet pretty quickly. She didn't really wait for me. Not like she had to, we are in college, but she never asked for my opinion. When we came the question where we had to use our probabilities to relate to a pond of 300 fish, she saw it easiest to convert to a percentage. That threw me off for a second because that seemed like and extra step that I hadn't thought of. My brain thought that a proportion would fit best for the scenario. If I didn't have confidence in myself a student, this may have thrown me off and caused me to question myself which I slightly did. But I decided to go for it my way anyhow and we came up with almost the same answer. She came up with one less healthy fish in the pond and one more sick fish than I did. She rounded using percentages so it makes sense why our answers were slightly varied. 

This moment was very important to me as an educator because even though my partner and I were working the same problem, we had different strategies. That is perfectly okay unless you are testing for mastery of a certain technique which... if they come up with the same answer, they shouldn't necessarily be counted wrong if they can explain their though processes. Every person learns and works differently. 

Note to self: This is extremely important to keep in mind to anticipate when are where work in students may vary and how specific you need to be in your directions. 

Another note to self: If you don't want students to eat the goldfish, tell them other students have also used these goldfish and put their hands ALLLL over them already :)