Depth · one concept from first principles · step 2 of 4
You roll a standard die with faces showing through . Among the following, find the one with the lowest probability.
The die can land on 1 through 6, all equally likely, so every chance is a count out of 6.
Listing every face first guarantees I never miss or double-count an outcome.
7.SP.C.7Make A Systematic ListEach of these chances is a count of faces out of the six equally likely faces.
All six faces are equally likely, so the whole certainty splits into six equal shares, one share sitting on each face.
An event's chance is then simply how many of those equal shares belong to it.
Each face that makes the event happen carries exactly one share, so counting the faces is the same as counting the shares.
The even faces are 2, 4, and 6 — three of them. So the probability of an even number is 3 out of 6.
Half the faces are even, matching the everyday sense that even is a 50-50 result.
7.SP.C.7Make A Systematic ListThe faces that are multiples of 3 are 3 and 6 — two of them. So this probability is 2 out of 6.
Only two faces fit, so this event is rarer than the even one.
7.SP.C.7Make A Systematic ListThe only face that is 6 or greater is 6 itself — just one face. So this probability is 1 out of 6.
There is nothing on a die above 6, so only the single face 6 counts.
7.SP.C.7Make A Systematic ListOver the same denominator 6 the numerators are 3, 2 and 1, so the last event is least likely.
With a common denominator, comparing probabilities is just comparing how many faces succeed.
7.SP.C.5Guess And CheckCount how many faces win, put it over 6, and the event with the fewest winning faces is the least likely.
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The same shape shows up later on the AMC, as Branching State Enumeration. Why the shapes repeat →