Probabilities less than 1: Guess the Misconception
It’s Guess the Misconception time again, and this week we turn our attention to one of my favourite topics — Probability.
Now, one of the golden rules of probability is, of course, that probabilities must be less than or equal to 1. How many times have we told our students that? So, what could possibly go wrong when students are faced with the following question?
Quite a lot, as it turn out.
I challenged you, as ever, to identify the most common incorrect answer to this question. Here are the results:
How on earth did only half the students answering this question (and it has been answered over 4,000 times) get it correct??? Let’s take a look at some student explanations to find out.
Answer A
Only 11% of students went for 2/3, but that could still be 2 or 3 students in our class. Students choosing this clearly do not think probabilities can be represented by fractions:
I think A because a probability has has to be a integer so 2/3 is a fraction so it cannot be a probability
I think it would be this one because a probability always adds up to 1 so I don’t think it would work
You don’t know what the value of 2/3 is whereas the other numbers give an appropriate probability
Answer B
This was the most popular choice of wrong answer, and there were some cracking explanations given:
Because it wouldn’t work in a probability as it doesn’t make a number like 1, 2, 3 etc unless rounded to the nearest whole number
I think B because its just a massive decimal and the rest look pretty legit. i also dont see how a number that big could be correct :I
I think it may not show the probability because the number to me doesn’t really look like it could be a probability
I find this so interesting, and it makes me think of the probability examples I use with my students. How often are they strange looking numbers like this? Not that often. They tend to come out as nice looking fractions, or decimals to at most two decimal places. Students thus form the implicit understanding that probabilities are always “nice” numbers. By not presenting students with a full range of all the different types of probabilities they may encounter soon into learning about the concept, it is not surprising that they come a cropper when they do meet something unusual.
Answer D
Again, we have the presence of a number that students have never associated with a probability before, and it leads to confusion:
D because its not a big number its to small so I would not be a probability of a something.
I think this because you wouldn’t see this in a probability question.
I think this because you cant have a 0.002 as a answer because it is too low.
Correct answer
Around half the students worked out the correct answer to this question, and here are some of the reasons they gave:
I think this is the answer because the probability scale stops at 1 and this number pass out the number 1.
A value of 1 equates to certainty, so a value of greater than 1 cannot possibly represent a probability — it is meaningless for something to be more than certain.
I would show 100% probability of something happening and you can’t go over 100% so 1.46 wouldn’t work
So, what are we to do? Well, my key takeaway is to ensure that the probabilities students encounter are not always the nice looking ones. Likewise, it is important that they experience nice looking numbers that cannot be probabilities. It is only through the presentation of such carefully chosen examples and non-examples that students will form a deeper, more complete understanding of the rule that probabilities must be less than or equal to 1.
So, I may present my students with an exercise like this:
I like to challenge students to vote whether each question could represent a probability or not. Fascinating debates and discussions usually ensue, and hopefully a more complete understanding of probability should emerge.
Why not try the diagnostic question out on your students, either in class or as part of a homework, and see how they get on? Talk about the correct answer, and also the wrong ones. Better still, you can ensure students receive a regular diet of quality questions like this — together with all the teacher insights you can ever want-by setting up our free schemes of work. We have free maths schemes from Year 1 to GCSE, with all the awarding bodies represented. Just click here to get started.
Have a great week
Craig
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