Common misconceptions in high school probability curriculum

Question

I am teaching probability to high school students. The material we are covering is pretty standard and includes:

• Introducing how to calculate the probability of events, e.g. coin flips, card draws, lottery tickets.

• The sum of the probabilities of all events is $1$

• Independent Events: $P(B \mid A) = P(B)$ and $P(A) \cdot P(B) =P(A \cap B)$

• Venn Diagrams and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Conditional probability: $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$

• Bayes Rule: $P(B \mid A) \cdot P(A) = P(A \mid B) \cdot P(B)$

• Product Rule: $P(B \cap A) = P(A) \cdot P(B \mid A)$

• Total Probability: $P(B) = P(A) \cdot P(B \mid A) + P(\bar{A}) \cdot P(B \mid\bar{A})$

• Binomial Trials

I am looking for examples of common misconceptions students have, or common mistakes that they do, when solving problems. Trivial examples are welcome. A preferred answer should also suggest ways in which one can prevent students to have said misconception.

An example of a common mistake would be:

Problem: You draw a card from a deck. What is the probability that it is red and has value J,Q,K or A?

Answer: $\frac{26}{52} + \frac{16}{52}$

My best suggestion for explaining to a student that this is wrong might be to change the problem to say red and black instead of simply red, and thus following the same logic getting a probability greater than $1$. But having students make similar thoughts seems unrealistic to me, and I am lost at how to best proceed.

Answer 1

Here are some things I occasionally encounter in the first few tutorial sessions as a TA for an undergraduate introduction to probability theory/statistics course.

• Why "and" corresponds to intersection, while "or" corresponds to union. Also, why "not" corresponds to the complement.
• The difference between $P(A \cap B)$ and $P(A \mid B)$, and which natural language statements to translate into which.
• That $P(A \cap B) = P(A)P(B)$ holds only for independent events $A,B$, while $P(A \cap B) = P(A\mid B)P(B)$ always holds; and that this corresponds to the fact that $P(A \mid B) = P(A)$ is another way to define independence.
• The fact that if $A = A_1 \sqcup A_2 \sqcup \cdots \sqcup A_n$, then $P(A) = \sum_{i=1}^n P(A_i)$.

Most of them would probably give the right answer if presented with the question explicitly and directly, but many of them made errors in the context of larger problems, or failed to think of the rule when it was natural to use it to solve a particular problem.

Answer 2

Regarding your example, a tree diagram might help to explain it properly. Also, basic stochastics often comes down to counting problems, that is we have to rephrase the problem such that every event gets the same probability (in your example, we are looking for four eight events out of 52, all having the same probability). If you can make your students understand this, you will help them a lot.

For an other example, take the question: "Throwing two dices, what is the chance of having the same number on each of them?". To properly solve it and more importantly develop the corresponding theory, one should look at all $36$ possible outcomes and from there deduce that the chance is $6/36 = 1/6$. There are of course other ways to see it (e.g. "fix one dice, what is the chance that the other gives the same?"), but in my opinion, turning it into a Laplace experiment is the one that can be used most often in such problems.

If you have control over the exercises, make your students always write down the total set of possible outcomes (maybe paraphrased if too big) and the set they are interested in.

Btw, a little advice on your way of explaining it with the probability bigger than one: This helps the student to see that there is something wrong there. Depending on their level of understanding, they will either say "ok, stupid rule with the one, whatever" or they will agree and understand that that can not be, however, they will still not be any wiser. You can use such an example with a very intelligent student to show him that he is wrong, but the general "I fu***** hate math!" student might just get demotivated by "BAM, you are wrong!", so a constructive approach to show them how to properly do it might be better. (I know I am exaggerating here and you most likely don't do it like that.^^)

A nice thing my own high school teacher did was the following: Every Friday, he gave us two different experiments (with dices, drawing balls,...). Then he left the room and gave us about 10min to discuss them, we where to compute the probability. Then he came back and we did the experiment we, as a class, chose. If we "won" - that is, we succeeded for example to through two times the same number with dices, we got no homework for the weekend. The chances for success where always between 10% and 30%, so we really had to figure out which one to choose if we wanted good chances. :) Of course for something like that, you need to know your class, you need the people to be motivated and you might not want to leave 30+ students alone to discuss (in this case maybe a majority vote for one of the systems might be better), but having such active experiments might always help.

Another nice thing he did (now I am really starting to remember^^): In the very first class, he wanted to show us that we had a natural understanding for Laplace experiments, all outcomes having the same probability. So he brought a dice, said we will through it 100 times and we should assume how often we will see a one. Turns out, we had 92x a one in the end, because it was a wooden dice and he filled the six holes opposite to the one with some heavy metal... xD However, we afterwards discussed why we, without knowing anything about the dice, assumed to see about $100/6$ ones and this was a great and fun start into the topic.

But to come back to your question of other examples: I think the most difficult thing is to find the right model for a problem. Most errors I myself made or encountered were of the form "damn, I should have counted that without repetition, with order,...". So I claim that the most difficult thing in basic stochastics is not to do counting, not to compute binomial coefficients but rather to translate an exercise into a model. It is also often not clear from the text if they are considering repetitions, etc., so you might want to make sure that your students not only write down computations or results but also a little explanation on which model they chose and why.

Answer 3

Very elementary issues that can hinder later understanding:

• Misunderstanding the difference between independent and mutually excusive and how to apply them.

• Not grasping that most elementary discrete probability problems are based on the modelling assumption that the probability of each outcome in a sample space are equal, and equal in value to $\frac{1}{size\,of\,sample\,space}$. Conversely a real-world (discrete) problem may have outcomes of non-equal probability - and how to handle that situation.

• Misunderstanding the definition of event in terms of outcomes and the definition of the probability of an event.

• Being unable to model a hypothesis test based of (an often poorly-worded) problem and not understanding the meaning of the result of the h-test.

• Grasping the difference between the frequency distribution of discrete outcomes/events and calculating an experimental/empirical probability and a probability calculated from a theoretical model - and how this can be one way into hypothesis tests. For example what is the probability of rolling a four on a fair 6-faced die if the die is rolled 600 times come shows a four 95 times? I don't think this is really a student misunderstanding - I just think it is a seemingly simple issue that is difficult to tackle at the start - which is when such scenarios are used to introduced probability.

Answer 4

In the actual teaching:

Definitely, independence of more than 2 events or random variables. Even teachers or professors make this mistake.

Independence of 3 events means pairwise independence and

$$P(\bigcap_{i=1}^3 A_i) = \prod_{i=1}^{3}P(A_i).$$

Few students think it's only pairwise. Many will think it's only the latter.

https://en.wikipedia.org/wiki/Independence_(probability_theory)#cite_note-4 http://www.engr.mun.ca/~ggeorge/MathGaz04.pdf

In the applications of probability or statistics in real life:

Nassim Nicholas Taleb aka Nero

Answer 5

A very common example (but not mentioned yet) is the birthday problem. I think this is a nice teaching example since it is so counter-intuitive. Also because it is easy to demonstrate (right in class) if you have a decent sized section.

See: http://www.math.cornell.edu/~mec/2008-2009/TianyiZheng/Birthday.html

P.s. I don't have my textbook any more (darned book borrower/thieves) but I would think your assigned textbook or alternate texts would contain several examples. (Not meant to deflect the question, but just that we don't have to solve this entirely from scratch. And/or that it might be useful to know in what ways the written examples are inadequate to guide further ideation.)