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I am hoping that someone can help me understand the standard 7.SP.7a, “Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of outcomes.” The comments in the draft progressions begin on the bottom of page 7 and continue to the top of page 8.
In the progressions document, it gives the example of selecting a given student’s name if there are 10 students as 1/10. What if there were two students who were named John, for example? I can assume that the example was talking about selecting each student as 1/10. If so, I understand that it is uniform because each student has a likely chance of being selected. The next sentence says, “If there are exactly four seventh graders on the list, the chance of selecting a seventh grader’s name is 0.40.” For the sake of argument let’s say there are 4 seventh graders, 5 sixth graders, and 1 eighth grader.
Here are my questions about that:
If we select one student at random from the ten, is that a uniform probability model regardless of the characteristic we note about that student (i.e. his/her name or grade level or color of their shirt or shoe size)? Or, does the characteristic we note about that student change the model from being uniform to not uniform.
That is, if all 10 students have different names and we choose 1 student out of 10 so that each name has a 1/10 probability, it is uniform. I think that’s easy to see.
But, if the students have the grade levels mentioned above and we choose 1 student out of 10 so that P(7th grader) is different that P(8th grader) is different, is the model now not uniform? Or is it still uniform because each student is equally likely to be chosen?
Also, is the P(7th grader event) considered a simple event or a compound event?
I would sincerely appreciate any clarification that you can offer. Thank you for your time.
You had me worried for a moment there that we had a major blunder in the standards, until I checked and realized you had misquoted the standard in an important way. It actually says “Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.” Can you see the difference? There’s a difference between outcomes and events; an outcome is an element of the probability space (in this case, a name on a list). In a uniform probably model, every outcome is assigned the same probability. So we assign every name on the list a probability of 1/10.
An event is a subset of the probability space; for example, the subset consisting of all students on the list who are in Grade 7. We can use the uniform probability model to calculate the probability of an event by counting the number of outcomes in the event (the number of students in Grade 7) and multiplying the probability of a single outcome.
As for your question about repeated names, I am imagining a list of full names, like a roll. Of course, two students could still have the same name, in which case presumably the school would have a way of distinguishing them. Still, we should make that clear.
Thanks for taking the time to reply. I do appreciate it very much. My sincerest apologies for misquoting outcomes and events in my post.
Is P(7th grader) compound?
I am curious to the answer to Steve’s question.I am a bit confused by the following terms, and I find disagreements in definitions in resources:
1. uniform probability model vs. uniform distribution – are these the same?
2. compound events vs. multi-stage events
According to the NCTM Navigating through Probability 6-8 book, a compound event is “an event that consists of more than one outcome”. Two examples given are P(one heads and one tails) is compound because there two outcomes from the sample space for one heads and one tails, and P(rolling a prime number).
However, in our textbook CCSS supplemental materials a compound event is defined as “an event that consists of two or more single events”. Is this describing a multi-stage event? i.e. flip a coin, spin a spinner.I am wondering how the term “compound event” is to be interpreted in SP.8? As more than one outcome, or more than one event (multi-stage)?
Thank you very much for your help!
Yes, a probability model and a probability distribution are the same thing, or at least two ways of looking at the same thing. A probability model assigns a probability to every event in the sample space. One way to conceptualize this (later in high school) is through the idea of a distribution, a function on the sample space for which the area under the graph above a certain event represents the probability of that event. (Sorry, compressed a large part of a course into that sentence!) And so, a uniform probability model and a uniform distribution are the same thing (note, however, that we also use the word distribution to refer to a data distribution, a different but related thing … but that’s another story).
The second definition of compound event (from your textbook) is the one used in CCSS. A compound event is an event in a sample space that has been constructed out of two other sample spaces. For example, you have the sample space {heads, tails} for tossing a coin and the sample space {1, 2, 3, 4, 5, 6} for rolling a die, and you construct the space
{(heads, 1), (heads, 2), (heads, 3), (heads, 4), (heads, 5), (heads, 6),
(tails, 1), (tails, 2), (tails, 3), (tails, 4), (tails, 5), (tails, 6)}out of both spaces, and calculate probability of events like “flip heads and roll an even number”, which is the subset {(heads, 2), (heads, 4), (heads, 6)}.
And yes, you might call this particular compound event a multi-stage event too. Although I can imagine cases where the two parts of the compound event happen at the same time (e.g. lightning strikes the tree and I am standing under it).
Here are a few more comments on “multistage” and “compound.”
I did a google search on “multistage event”. I don’t get a lot of hits related to probability and statistics for that but I do for “multistage experiment”, which gives the term a slightly different emphasis. An experiment is something like tossing a coin, tossing two coins, rolling a die, etc. It’s what you do to generate an outcome.
“Compound event” appears to have two definitions which are not equivalent. Sometimes the idea that there are two different definitions of a term comes as a shock, but it does happen. Trapezoid is one example.
The NCTM book Navigating Through Probability, 6–8 says on p. 11: “An event is the outcome of a trial. A simple event (usually called an event) is a single outcome. A compound event is an event that consists of more than one outcome.” For the experiment “roll a die,” it gives the example of “six on top face” for simple event, “prime number on top face” for compound event. Neither of these is the result of a multistage experiment (“roll a die” has only one stage).
That’s the definition of compound event that I grew up with. Using that definition and using the sample space described above (i.e., there are four 7th graders on the list), the event “picking a 7th grader” is a compound event and “picking an 8th grader” is a simple event.
In the CCSS, “compound event” is more akin to “an outcome of a multistage experiment,” e.g., an outcome of rolling two dice, as Bill has already discussed. Under that definition, “picking a 7th grader” as described above is not a compound event. The experiment is “picking a student” which has only one stage.
I can see that we need a note about this in the S&P Progression.
I am trying to qualify uniform probability models with my colleagues and I need to clarify some of your discussion last year with SteveG. The reason that selecting a student is a uniform probability model is because each of the 10 students (the outcomes) has an equal chance of being selected, correct?
In SteveG’s question, he moved from the experiment (selecting a student) to a specific event – select a 7th grader (a subset of the sample space). The selection of a student is equally likely, but the selection of a 7th grader is not equally likely (0.4 to 0.6), but each student is still equally likely to be chosen.
Let’s say we have a spinner divided into equal fifths. If the sections were labeled 1 to 5, then the spinner would be a uniform probability model because the sample space is {1, 2, 3, 4, 5} and each outcome is equally likely to happen. Spinning an odd number would be the sum of all the probabilities of the odd numbers = 1/5 +1/5 + 1/5 = 3/5. Just like above, the spinner is equally likely to land on any number on the spinner.
Taking this one step further, take the same spinner, but label two sections “1” and three sections “2.” This is no longer a uniform probability distribution because the sample space is now {1, 2}, but P(1) = 2/5 and P(2) = 3/5. The outcomes no longer are equally likely to occur. Would you agree with this?
Yes, this sounds basically right. Although I wouldn’t call the spinner a model, but rather a representation of the model. The model itself is just the sample space {1, 2} with the assigned probabilities P(1) = 2/5 and P(2) = 3/5.
Not sure this is the correct place for this question but as a follow up to the above question and using the spinner described would it be appropriate in the middle grades to ask a student to determine the probability of spinning a 1 followed by a 2? or the probability of spinning and getting a 1 on both spins? Could this be presented using a tree diagram with the probabilities given and find the probability of each possible event given that the spinner is used 2 times? Or is this included in the high school standards for probability?
Yes, using a tree diagram for this is entirely consistent with 7.SP.8. The main thing would be not to get hung up on teaching general rules for calculating the probability of compound events—that really is high school, as in S-CP.B—but rather to give students concrete experiences that prepare them for those abstract rules.
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