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All courses Math Foundations · MF.DP.4 58 of 60
Solve simple compound probability and counting problems using organized reasoning.

Count outcomes in a two-step situation

Problem
A coin is flipped \(\text{once}\), and then a standard number cube with \(6~\text{sides}\) is rolled. How many total outcomes are possible?
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Hint

Write the coin results first: H and T. Then ask, “How many number-cube results can go with H? How many can go with T?”

This is a two-step experiment, so each coin result can pair with every one of the 6 cube results. You can list the pairs or use multiplication.

Solution walkthrough

01

Count the first experiment's outcomes

\[\text{coin}:~{H,T}~->~2~\text{outcomes}\]

A single coin flip has two possible results.

02

Count the second experiment's outcomes

\[\text{number}~\text{cube}:~{1,2,3,4,5,6}~->~6~\text{outcomes}\]

The standard cube has six possible results.

03

Apply the multiplication principle

\[\text{total}~\text{ordered}~\text{outcomes}=2\times~6=12\]

Each of the two coin results can pair with every one of the six cube results.

04

Check the sample-space structure

\[H_{1}–H_{6}:6~\text{outcomes};~T_{1}–T_{6}:6~\text{outcomes};~6+6=12\]

Listing by coin result confirms that no pair is missed or repeated.

05

Report the total

\[12\]

There are 12 possible outcomes for the two-stage experiment.

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Another way

  1. Draw a two-branch tree for H and T, with six number branches from each, giving twelve leaves.

!

Common mistake

Do not add 2 and 6; compound choices pair, so use the multiplication principle.

Solution walkthrough video