Main content

### Course: Integrated math 2 > Unit 13

Lesson 1: Probability basics- Intro to theoretical probability
- Probability: the basics
- Simple probability: yellow marble
- Simple probability: non-blue marble
- Simple probability
- Experimental probability
- Experimental probability
- Theoretical and experimental probabilities
- Making predictions with probability
- Making predictions with probability
- Intuitive sense of probabilities
- Comparing probabilities
- Probability models example: frozen yogurt
- Probability models
- Theoretical and experimental probability: Coin flips and die rolls

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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?(15 votes)
- 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.(42 votes)

- 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."(24 votes)
- Sometimes that may happen. Or, to phrase that better, the chances of it happening are less, but it can still happen. If you were to flip 10 coins, you might get numbers such as 4 heads and 6 tails. But the higher the number of times you flip the coin, the closer both numbers will be to exactly half.(3 votes)

- 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!(16 votes)

- 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?(7 votes)
- 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!(7 votes)
- What would be a good definition of experimental probability?(3 votes)
- 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).(8 votes)

- Why isn't it 7,000 and 3,000? Not 8,000 and 2,000

Sorry this is probably a dumb question(4 votes)- 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.(6 votes)

- how can there be 10,000 experiments if there are only 100 experiments in all to be taken?(2 votes)
- Every time after a marble is drawn, it is put back in the bag, so there are always 100 marbles when you draw,(10 votes)

- Are the experimental probabilities closer to the theoretical probability?(3 votes)
- Would the 80% of 10,000 still be a problem if they were tiny spheres and a bag contained a million?(4 votes)
- Yes, the problem after 10,000 tests would be about equally large if the bag held a million marbles and we assumed that there were exactly 500,000 of each, or if the bag contained just two marbles and we assumed that there was one of each.(1 vote)

## Video transcript

- Let's say that you've got a bag, and in that bag you
put a bunch of marbles. So, let's say you put 50
of these magenta marbles. So one, two, three,
four, five, six, seven, I'm not gonna draw all of them
but you get the general idea. There are going to be 50 magenta marbles, and there's also going
to be 50 blue marbles. And what you do is, you have
these 100 marbles in there, half of them magenta, half of them blue. And before picking a marble out, and you're gonna be blindfolded when you pick a marble out, you shake the bag really good to, so you think, mix them up a little bit. And so, if you were to say theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability
of picking a magenta? I feel the need to write
magenta in magenta. What is the probability of
picking a magenta marble? Well, theoretically there's 100 equally likely possibilities, there's 100 marbles in the bag. And 50 of them involve picking a magenta. So, 50 out of 100, when this is the same thing as a 1/2 probability. So you could say, well, "Theoretically, "there is a 1/2 probability,
I just did the math." If you say these are 100
equally likely possibilities, 50 of them are picking magenta. Now let's say you actually
start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what
marble color you picked, and you put it back in,
and then you do it again. And so, let's say that after every time you put your hand in the bag and you take something out of the bag, and you observe what it is, we're gonna call that an experiment. So, after 10 experiments, let's say that you you have picked out seven magenta and three blue. So, is this strange that out
of the first 10 experiments, you haven't picked out exactly
half of them being magenta, you've picked out seven magenta, and then the other three were blue. Well no, this is definitely
a reasonable thing. If the true probability of
picking out a magenta is 1/2, it's definitely possible
that you could still pick out seven magenta, that just happened to be what your fingers touched. And this isn't a lot of experiments, it's completely reasonable that out of 10, yeah, you could have,
later on in statistics we'll define these things in more detail, but there's enough variation in where you might pick that you're
not going to always get, especially with only 10 experiments, you're not definitely
going to get exactly 1/2. Instead of having five magenta, it's completely reasonable
to have seven magenta. So, this really wouldn't
cause me a lot of pause. I still wouldn't question what I did here when I calculated this
theoretical probability. But let's say you have a
lot of time on your hands. And let's say after 10,000 trials here, after 10,000 experiments,
and remember the experiment; you're sticking your hand
in the bag without looking, your fingers kind of feeling
around, picks out a marble, and you observe the marble
and you record what you found. And so, let's say after
10,000 experiments, you get 7,000 magenta. Actually I'm gonna do
slightly different numbers, so let me make it even more extreme. Let's say you get 8,000 magenta and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different. And now you have a large number of trials right over here, not just 10. 10 is completely reasonable that, hey you know, I got seven magenta and three blue instead of five and five, but now you've done 10,000. You would've expected if this was the true probability,
you would've expected that half of these would've been magenta, only 5,000 magenta and 5,000 blue, but you got 8,000 magenta. Now, this is within the
realm of possibility if the true probability of
picking a magenta is 1/2, but it's very unlikely
that you would've gotten this result with this many experiments, this many trials if the
true probability was 1/2. Here your experimental
probability is showing, look, out of 10,000 trials, experimental probability here is you had 10,000 trials,
or 10,000 experiments I guess you could say. And and in 8,000 of them,
you got a magenta marble. And so, this is going to be 80%, or 8/10. So, there seems to be a difference here. The reason why I would
take this more seriously is that you had a lot of trials here, you did this 10,000 times. If the true probability was one half, it's very low likelihood that you would've gotten this many magenta. So, when you think about it you're like, "What's going on here, what are "possible explanations for this?" This, I wouldn't have fretted about, after 10 experiments, not a big deal. But after 10,000, this
would have caused me pause. Well, why would this happen, I mixed up the bag every time. And there're some different possibilities; maybe the blue marbles
are slightly heavier, and so when you shake the bag up enough, the blue marbles settled to the bottom, and you're more likely
to pick a magenta marble. Maybe the blue marbles
have a slightly different texture to them, in which case, maybe they slip out of your hands,
or they're less likely to be gripped on, and so you're more likely to pick a magenta. So, I don't know the explanation, I don't know what's going on in that bag, but if I thought theoretically that the probability should be 1/2, because half of the marbles are magenta, but I'm seeing through my experiments that 80% of what I'm picking out, especially if I did 10,000 of them, if I did this 10,000 times, well, this is going to cause me some pause. I would really start to think about whether it's truly equally likely for me to pick out a red, a
magenta versus a blue. Something else must be going on.