Experimental vs. Theoretical Probability

Learning Targets Difference between experimental vs. theoretical probabilities. How to record the probabilities and make inferences based on results.

Definitions Experimental Probability: is the ratio of the number of times an event occurs to the total number of trials or times the activity is performed. Theoretical Probability: is finding the probability of events that come from a sample space of known likely outcomes.

Experimental Probability In order to find the experimental probability of an event we must actually “perform” the task. The experimental probability may or may not mimic the actual odds we should expect. Our sample space is limited to the number of trials ran.

Theoretical Probability In order to determine the theoretical probability we have to look at all of the possible outcomes and determine which ones are desired The desired outcomes divided by all possible outcomes will give the probability of that event happening in the long run. This means that by repeating the action/game/experiment for long periods of time our outcomes will mimic the expected probability.

Experimental vs. Theoretical You decide to roll a dice and flip a coin 40 times. Here are the results of the experiment: Dice Number Rolled Coin Flip Number of Times 1 H 13 2 T 4 16 5

Experimental vs. Theoretical What is the experimental probability of rolling a 1 and flipping a heads? What is the experimental probability of at least rolling a 2? Flipping a tails? What is the theoretical probability of these events? How would you change the experiment so that you get your probability closer to the theoretical probability? Dice Number Rolled Coin Flip Number of Times 1 H 13 2 T 4 16 5

Probability Both methods are valid ways of determining outcomes. It is important to look at all of the facts when judging whether outcomes are normal. Factors that can impact outcomes: Number of trials Validity of experiment (biased, random, etc.) Reporting/recording errors

Why do we care about Experimental Probability? Experimental probability seems less important given that theoretical probability will give us the actual odds. Some events are just too big and impossible to count all of the outcomes for. It is easy to plot the sample space for a deck of cards, flipping a coin or spinning a spinner.

Using Experimental Probability Have you ever wondered… How do political ratings and surveys manage to get such accurate results without surveying every person in the country? The amount of fish in a lake? Medicinal side effects?

Using Experimental Probability By using experimental probability we can determine the likelihood of events happening They key is to take enough samples and record them properly in order to make inferences Example: Gallup Poll Surveys

Gallup Poll The Gallup Poll uses experimental probability to determine the views, odds or likelihoods of the nation. They randomly survey a 1,000 adults throughout the country via telephone interviews. This random sampling allows them to get data and make inferences on the larger population

Fish in Crater Lake

Crater Lake As a marine biologist working for the State of Oregon you are asked to see if the population of trout in Crater Lake is still healthy and thriving. You decide to go out and tag some trout for counting. All day you catch and release large quantities of fish but tagging all of the trout you come across.

Crater Lake At the end of the day you have tagged 316 trout. Over the next few weeks you go out and repeat the catching and releasing of fish at random areas and times. You count the number of trout caught, but also count the number of tagged trout.

Crater Lake Sample Number Total Trout Caught Number of Tagged Trout 1 457 23 2 643 45 3 511 20 4 756 49 5 427 22 6 598 30

Crater Lake Based on our sampling how many trout are there in Crater Lake? Explain your reasoning…

Medical Trials The FDA is testing out your new pharmaceutical drug. It is in the final phases of testing by giving trial doses to humans. Out of the real doses given to patients the following data is given:

Medical Trial Results Trial Patients Tested Side Effect: Stomach Bleeding High Blood Pressure Stroke 1 3,023 45 696 2 2,595 44 636 3 4,389 75 1,317 4 6,784 81 1,696 5 3,658 55 841 6 5,008 70 1,177

Medical Trials If this drug is expected to be given out to 20 million people how many people should you expect to experience the following side effects? Stomach Bleeding? High Blood Pressure? Stroke?