This is Common Core State Standards Support Video in mathematics. The standard is 7.SP.7a. The standard states: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain sources of the discrepancy. Part A states: Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probability of events.
The key ideas or concepts for this standard—well first, the standard focuses on simple events. There’s a subsequent standard 7.SP.8 that deals with compound events. So the focus here is probability dealing with simple events. This terminology isn’t used in the standard, but theoretical probability is probability based on math theory. It’s the expectation of what should happen in a probability context or experiment. Experimental probability represents the actual results obtained from performing probability experiments. Now this last item is important. It’s not mentioned in the standard, but it’s something that’s a common idea in probability. And that is that the more trials or events we carry out, the closer the experimental probability is to the theoretical probability. In other words, the more trials that you do, what actually did happen will get closer and closer to what should have happened.
Okay, let’s look at a simple probability context, that of flipping a coin. Now of course, the expectation would be that a coin toss would result in heads and that would be 1/2 or .5 or 50%. Now a footnote here is that this is a good opportunity to review or teach conversions among fractions, decimals, and percent. Those topics are addressed in standards 4.NF.6 and 6.RP.3c. So again, these contexts are a good way to go back and either teach or review the conversions, because it is important that students know how to convert among fractions, decimals, and percent.
You might have some software that has something like this where it will actually flip a coin or of course, you could just do it physically with real coins. Let’s say you flip the coin, and of course the expectation would be that heads should come up 50% of the time. But your actual results, you flip the coin 4 times, and 3 out of the 4 came out to be heads which is 75%. So it isn’t what you expected. You expected it to be 50% instead of 75%. Now one of the parts of the standard states that you need to compare probabilities from a model to observed frequencies, and if the agreement is not good, explain the sources of the discrepancy. So here, a logical explanation would be that we just haven’t done enough trials. We only did 4. In fact if you stop and think about it, here all that had to happen for the results to be what we would have expected would have been if one of these heads would have turned out to be tails. Then we would have had exactly, 50%. But that’s not what actually happened.
So here again, the logical explanation is that we just didn’t do enough trials. Now let’s say, we took the same idea, and we did an experiment where we flipped a coin 100 times, and of course initially the data would have been recorded with tally marks. Then when they counted up all the marks, this is the result—we got 46 heads and 54 tails. So the actual result for heads was 46%. Notice that that 46% is closer to what we would’ve expected, the 50%. So again, this is one of those cases where the more trials that you do, what actually happens will get closer and closer to what should happen.
Okay, let’s take a spinning wheel. Again you might be able to find this somewhere online or maybe you have software that has something like this where again you can take the spinning wheel and spin it, and of course, the expectation is that there’s an equal probability that you could land on blue, red, green, or white. Now an important idea here would be well, why? Why is it equal probabilities that it could land on white, red, green, or blue? Well notice that each of these parts is the same size, so that’s the connection. That’s how come you have an equal probability. If these were different sizes, then that would change the context. It would not be equal probabilities of each one of those happening.
So let’s say we do a trial, and we’re looking at the blue. When we do our tally, we see that out of the 12 trials, the 12 spins that we did, 5 of those instances resulted in blue, which comes out to be 42%, which is different than our expectation of 25%. Now that wasn’t very many trials; that was only 12. So let’s say we do the experiment but with a lot more spins, let’s say 200. And this time, out of the 200 spins, 52 times it came out blue. When we do the computation, that’s 26%, which is a lot closer to the 25% that we expected.
Now let’s say we would have done that same experiment and we would have done 200 spins. But let’s say this was our result, we got blue again. We got blue 86 times out of 200, which is 43%. Now there’s something wrong here; this is not what should have happened. Now if you were using a physical spinning wheel, one possibility would be that maybe there’s a raised part or something where maybe the arrows are bumping it, and it slows it down to where it’s more likely to hit on blue or maybe roll a little bit over to white. So again, that’s just one possibility, that we did a whole lot more trials, but the expected results didn’t quite match what really happened. So again, there’s got to be some reason. In this case, it was something other than not doing enough trials.
Let’s take another context. Let’s say you have a game show. You have a grand prize, and it can be behind door number 1, door number 2, or door number 3. Now the expectation would be that there’s an equal amount of chance that a contestant is going to choose one of those three doors—1 out of 3 for door 1, 1 out of 3 for door 2, and 1 out of 3 for door 3. But let’s say that the producers of the show keep up with some statistics, and let’s say that after 99 shows this is the result——29 out of the 99 contestants picked door number 1, 44 out of the 99 picked door number 2, and 26 out of the 99 picked door number 3.
So for whatever reason here, (I guess it’s just human nature), for some reason the contestants tended to pick number 2 more than 1 or 3. Now the producers of the show might decide, well we need to make this more representative. We need more winners. So let’s start putting the prize behind number 2 a little bit more often. But what could also happen would be that maybe the producers decide we need to save some money. We don’t want to give out as many prizes. So let’s put the prize more often behind door number 1 and door number 3, because if this pattern continues, they’re more likely to pick number 2 and not win it.
In looking at the first part of the standard: Develop a probability model and use it to find probabilities of events, the expectation would be that a student can take a context and establish some type of probability experiment. They can determine the expected or the theoretical probability. They should perform the experiment and calculate the actual results and then compare the two; compare what really happened to what we expected to happen.
Now this scenario would be that okay, you’ve got a jar with different colored marbles. The experiment would involve reaching in without looking and randomly picking a marble and then seeing what it is. But then of course, you have to put it back; you have to replace it so that we have the same number to choose from every time. Now notice here that we have different numbers. If we’re going to adhere to strictly this standard, you’d want to make them all the same so that you have equal probability of each. So maybe you would make all of them 6—you’d have 6 blue, 6 red, 6 green, and 6 purple. But after a while, if students are up to the challenge, you could do something like this where you have different amounts of each. So then you’re going to have different expected probabilities. So that would be the expectation in that kind of scenario where they would be expected to figure out the probability of picking each one of the four different colors, which in this case would be what is shown here.
Let’s say okay, they’ve done a trial, and again, the standard states that they need to compare probabilities from a model to observed frequencies. If the agreement is not good, explain sources of the discrepancy. So let’s look at the results here. If we focus on the green, twice out of the 12 times that we reached in to pick out a marble, it came out green, and that’s only 16.7%. That’s not the expectation. It was closer to 22.2%, kind of close but again, not quite what we expected. And the justification here would be that okay, we only did 12 trials. We really need to do more trials to see if we really do get closer and closer to what we would expect for each of these different percentages.