Students will be able to solve a variety of basic probability problems

Factorials, permutations and understanding independent and mutually exclusive events are critical to solving basic probability problems

60 minutes

I like to use this assessment to get a sense of how well my students are understanding the probability unit. Although it does seem summative, I use the data in a follow up lesson as described in the next section.

**source**: http://www.regentsprep.org/Regents/math/ALGEBRA/MultipleChoiceReview/Probability.htm

60 minutes

On the following day of the assessment, students enter and I hand them markers. I then give them the answers to the problems and have them grade themselves. Then I collect their markers and give them time to fix their work. At the end of class we review a few of their misconceptions.

Here are the answers: Answers

There are a few common misconceptions that come up with these problems. I will review three here.

**Question 1 (the gambler's fallacy question)**: What is the chance of flipping a head after x amount of trials.

This is one of my favorite types of probability questions, since many people believe if you haven't had your number come up yet, it must come up soon. However, the probability resets with each trial. This is the basic definition of an independent even and needs to be understood. If students aren't convinced, ask then to consider the absurdity of the opposite. If all the previous flips of a coin weighed into the current result, how fair would *any* toss be? As soon as a coin is physically turned that would skew the next result.

**Question 13: (repeat digits):** How many five letter arrangements can be formed from the ed word "digit."

Perhaps the best way to approach this question is to first encourage the students to pretend we actually have 5 distinct letters and do the calculation 5! = 120

Then ask, "how many of those will be repeats?" The answer is half, since two of the letters are repeats. So here we can solve with 5!/2

If they aren't convinced try a much smaller sample, something like the work Keen and show them all the 12 arrangements:

**KEEN**

**KENE**

**KNEE**

**EEKN**

**EENK**

**EKEN**

**EKNE**

**ENEK**

**ENKE**

**NEEK**

**NEKE**

**NKEE**

You can *double this list* if you color code the two E's and treat them as two different letters. Then students will see the 24 total combinations. However it is obvious that only half of them are truly different. It is difficult for students to think about why we don't treat the two letters differently but why we might treat something like colors differently. I like this conversation because we could color code the letters and make them different, but it important to understand that our analysis should fit the problem we are dealing with.

**Question 20: (Mutual Exclusivity)** A piggybank contains 2 quarters, 3 dimes, 4 nickels, and 5 pennies. One coin is removed at random. What is the probability that the coin is a dime or a nickel?

First identify the total number of coins: 14

Then list out the options:

3/14 chance of getting a dime or a 4/14 chance of getting a nickel

Students tend to multiply these two values (especially after doing all those permutations). To spiral back to our work with Venn Diagrams, we set this problem up in a Venn diagram and reexplain the idea of sets being mutually exclusive. They need to realize that you are increasing your success by broadening the types of things you want to get. If that approach doesn't work, I suggest using extremes to get the point across. Ask them to imagine the probability of getting a dime or nickel or penny or quarter. Would it make sense to multiply all of those probabilities? What happens if we add them? Why should we get a sum of 1?

Asking these questions will help students step back and make sense of probability questions.

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