Lesson #2 - Augsburg College

Lesson #8: Combinations
Approximate lesson length: 1-2 days
Precedes lesson #9
Fits well with lesson #10
Probability: What are the Chances?!
Lesson Objectives:
• Define event and outcomes
• Discover a way to count the total possible combinations of rolling a die twice or three times
• When rolling two dice, find the probability of rolling a 5 and then a 6
Materials:
• Pencil and paper
Begin by asking each student to write down in their own words the definition of probability. Ask everyone
to share and discuss their responses. Pose the question: What’s the probability of rolling a six when
you roll 1 die? And a four? Two? One? (It’s 1 out of 6!) Why? (Because there’s six possible numbers
and they’re all different!) Continue the discussion until you find students are making this comment confidently
and understand what the probability of rolling a given number 1-6 is and why. At this point tell
them that they already know the strategy for calculating any probability: Find out how many total possible
outcomes there are, then count how many of the possible outcomes count as the event.
Write this strategy on your own white board or paper as you talk it through with them. Ask the students
to describe in their own words what they think ‘outcomes’ means and what ‘event’ means. They
should be able to realize that in this case there are 6 total possible outcomes: rolling a 1, rolling a 2, rolling
a 3, rolling a 4, rolling a 5, and rolling a 6. The event “rolling a six” only happens in one of the outcomes.
Following the strategy, the probability of rolling a six is 1/6.
Once the group is comfortable conceptualizing how we calculate the probability of rolling any single
given number, pose this more difficult question: What is the probability of rolling a 5 on the first roll and
then a 6 on the second roll? Open up a discussion about how we might go about figuring this out.
To figure this out, we need to know how many possible combinations there are of rolling a die twice.
Let the ordered pairs (A,B) represent rolling A first and B second. So for example (1,1) means I rolled a 1
first and a 1 second. (2,4) means I rolled a 2 first and a 4 second. Students may come up with their own
notation for this concept, but if they don’t, suggesting this method may be helpful to them.
Ask the students to write down all the possible ways to roll a die twice. When I taught this lesson, every
student began by writing down lots of random combinations. This led me to ask the question: Is there
any way we can write down all of these combinations so that we can keep track of which ones we’ve accounted
for and which ones we still need? Ultimately, making a table like the following is what emerged:
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