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!