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## Grade 8 math (FL B.E.S.T.)

### Course: Grade 8 math (FL B.E.S.T.)>Unit 9

Lesson 4: Sample spaces & probability

# Theoretical and experimental probabilities

Compare expected probabilities to what really happens when we run experiments.

## Want to join the conversation?

• How can I tell the difference between experimental probability and theoretical probability? •  Experimental probability is the results of an experiment, let's say for the sake of an example marbles in a bag. Experimental probability would be drawing marbles out of the bag and recording the results. Theoretical probability is calculating the probability of it happening, not actually going out and experimenting. So, calculating the probability of drawing a red marble out of the bag.
• When my older brother was learning about probability, he flipped a coin to experiment, and when he flipped the coin 8 times 7/8 of the time he got tails. His teacher's reaction: "That coin is rigged." • Isn't the probability still 50% but it just so happens that you got in this experiment an 80%?
(1 vote) • That confused me too this year in seventh grade ;) As far as your question, you are totally right! Although the theoretical (expected) probability is 50%, the experimental probability doesn't have to be 50%. I like to picture a coin flip-- the theoretical probability is that the coin will land on heads once if you flip it two times, but it will not always land on heads once. Technically in your mentioned experiment, you could get any percentage even though the estimated percentage is 50%. Hope this is helpful!
• What would be a good definition of experimental probability? • The experimental probability of an event is an estimate of the theoretical (or true) probability, based on performing a number of repeated independent trials of an experiment, counting the number of times the desired event occurs, and finally dividing the number of times the event occurs by the number of trials of the experiment.

For example, if a fair die is rolled 20 times and the number 6 occurs 4 times, then the experimental probability of a 6 on a given roll of the die would be 4/20=1/5. Note that the theoretical probability of a 6 on a given roll would be 1/6, since it is given that the die is fair. So experimental probability can differ from theoretical probability.

As the number of trials in the experiment grows towards infinity, the experimental probability almost surely converges towards the theoretical probability (law of large numbers).
• So basically it's saying that there can be unknown variables or factors that may influence the probability, and also that a higher sample size is better for calculating probability, right? • That confused me too this year in seventh grade ;) As far as your question, you are totally right! Although the theoretical (expected) probability is 50%, the experimental probability doesn't have to be 50%. I like to picture a coin flip-- the theoretical probability is that the coin will land on heads once if you flip it two times, but it will not always land on heads once. Technically in your mentioned experiment, you could get any percentage even though the estimated percentage is 50%. Hope this is helpful! • Why isn't it 7,000 and 3,000? Not 8,000 and 2,000
Sorry this is probably a dumb question • The exact numbers don't really matter.
Sal's main point in the video is that when you conduct a lot of experiments, the experimental probability should get increasingly close to the theoretical probability (or in mathematical terms you could say the experimental probability should approach the theoretical probability). If it doesn't, then it's a signal to start questioning the results.

For instance , in this example the theoretical probability of drawing a specific color marble is 50%. So while it's totally probable to get a 70% or even 80% magenta marbles in a small number of experiments, these odds become increasingly unlikely with a larger number of experiments.

So, Sal intentionally used a large number like 8000 magenta marbles, rather than a smaller one, to demonstrate that there is something wrong with the experiment and that we should investigate the reason behind the experimental probability not approaching the theoretical one.   