Probability is defined as the ratio of the number of desired outcomes (what you want to happen) to the number of total possible outcomes (what could possibly happen). Probability is a ratio, so it can be expressed as a fraction, a decimal, or a percent.

In probability, a simple event is one action or observation. For example, you could roll a six-sided number cube. An event in this situation would be a number landing face up on the number cube. Face up means the number that is showing on the top face of the number cube.

• If each face is numbered with a digit from 1 through 6, what are the possible outcomes for rolling a number cube?

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  There are 6 outcomes: 1, 2, 3, 4, 5, or 6 could appear on the top face of the number cube.

Use the interactive below to see a net of a six-sided number cube and how the net can be folded to create a number cube.

Use the six-sided number cube shown in the animation to answer the following questions.

• What is the probability that Bernard will roll a 3 if he rolls the number cube once?

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  The desired outcome for this event is a 3. How many ways can a 3 land on the top face of the number cube?
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  P(3) is the probability of rolling a 3 face up on a number cube.
  
  P(3) = number of 3's over total number of outcomes Number of 3's Total Number of Outcomes = 1 over 6 1 6
• What is the probability that Shyanne will roll an odd number if she rolls the number cube once?

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  The desired outcome for this event is an odd number. How many ways can an odd number land on the top face of the number cube?
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  Odd numbers on the number cube include 1, 3, and 5.
  
  P(odd) = number of prime numbers over total number of outcomes Number of Odd Numbers Total Number of Outcomes = 3 over 6 3 6 = 1 over 2 1 2
• What is the probability that DeWayne will roll a prime number if he rolls the number cube once?

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  The desired outcome for this event is a prime number. How many ways can a prime number land on the top face of the number cube?
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  Prime numbers on the number cube include 2, 3, and 5.
  
  P(prime) = number of prime numbers over total number of outcomes Number of Prime Numbers Total Number of Outcomes = 3 over 6 3 6 = 1 over 2 1 2

Pause and Reflect

How can you determine the total number of outcomes for an event?

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Use the sample space to create a list of all possible outcomes.

How can you determine the number of desired outcomes for an event?

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Look in the sample space, and identify any outcomes that meet the criteria for the event. For example, if the desired outcome is a prime number, look for any prime numbers that appear in the sample space.

Practice

Source: 4d8, Troumad, Wikimedia Commons

Brianna and Desmond use an 8-sided die (which is not a cube, but an octahedron) to play a board game. Each face of the octahedron contains a digit from 1 to 8, and each face of the octahedron is congruent, so each number has an equal chance of appearing on the top face.

1. What is the probability that Brianna will roll an even number?

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   How many even numbers are there on an 8-sided die? How many total outcomes are there?
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   Even numbers on the octahedron include 2, 4, 6, and 8.
   
   P(even) = number of prime numbers over total number of outcomes Number of Even Numbers Total Number of Outcomes = 4 over 8 4 8 = 1 over 2 1 2
2. What is the probability that Desmond will roll a number less than 4?

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   How many numbers on the 8-sided die are less than but not equal to 4? How many total outcomes are there?
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   Numbers on the octahedron that are less than 4 include 1, 2, and 3.
   
   P(less than 4) = number of prime numbers over total number of outcomes Number of Desired Numbers Total Number of Outcomes = 3 over 8 3 8
3. Brianna will roll the octahedron next. Does Brianna have a greater chance of rolling a prime number or rolling a number greater than or equal to 6?

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   How many numbers on the 8-sided die are prime? How many numbers are greater than or equal to 6? How many total outcomes are there?
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   Prime numbers on the octahedron include 2, 3, 5, and 7.
   
   P(prime) = number of prime numbers over total number of outcomes Number of Prime Numbers Total Number of Outcomes = 4 over 8 4 8 = 1 over 2 1 2
   
   Numbers on the octahedron that are greater than or equal to 6 include 6, 7, and 8.
   
   P(greater than or equal to 6) = number of desired numbers over total number of outcomes Number of Desired Numbers Total Number of Outcomes = 3 over 8 3 8
   
   Since 1 over 2 1 2 is greater than 3 over 8 3 8 , Brianna has a greater chance of rolling a prime number than rolling a number greater than or equal to 6.