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!