Question

If a pair of dice is rolled, what is the probability that both will land on an odd number? A. \(\frac{1}{36}\) B. \(\frac{1}{12}\) C. \(\frac{1}{4}\) D. \(\frac{5}{6}\)

Short answer

\(\frac{1}{4}\)

Step-by-step solution

Identify the total number of outcomes

First, determine the total number of possible outcomes when rolling a pair of dice. Each die has 6 faces, so the total number of possible outcomes is given by the product of the number of faces on each die: \[6 \times 6 = 36\]

Count the favorable outcomes

Next, count the number of favorable outcomes where both dice land on an odd number. Each die has three odd numbers (1, 3, 5). The total number of favorable outcomes is the product of the number of odd numbers on each die: \[3 \times 3 = 9\]

Calculate the probability

To find the probability, divide the number of favorable outcomes by the total number of possible outcomes: \[\frac{9}{36} = \frac{1}{4}\]

Key concepts

outcomes in probability

When we talk about outcomes in probability, we mean all the possible results that can happen in an experiment. In this case, the experiment is rolling a pair of dice.
Each die has 6 faces with numbers from 1 to 6. When we roll two dice, each of the 6 faces of the first die can pair with each of the 6 faces of the second die.
So, the total number of outcomes when rolling two dice is given by multiplying the number of faces on the first die by the number of faces on the second die:
\[6 \times 6 = 36\]
This gives us our total outcomes: 36 different possibilities whenever we roll a pair of dice. Understanding the total number of outcomes is crucial before calculating probabilities.

favorable outcomes

Favorable outcomes refer to the specific results we are interested in within the set of all possible outcomes. In this exercise, we want to find the probability that both dice will land on an odd number.
Each die has three odd numbers: 1, 3, and 5. To find the favorable outcomes, we need to count how many ways two dice can show these odd numbers.
Each odd number on the first die can pair with each odd number on the second die. So, the total number of favorable outcomes will be:
\[3 \times 3 = 9\]
There are 9 pairs such as (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), and (5,5). These are the favorable outcomes where both dice show an odd number.

calculating probability

Calculating probability involves comparing the number of favorable outcomes to the total number of possible outcomes. In probability, this is usually expressed as a fraction.
We already know that:

• The total number of possible outcomes is 36.
• The number of favorable outcomes (both dice showing an odd number) is 9.

To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
\[\frac{9}{36} = \frac{1}{4}\]
This means the probability that both dice will land on an odd number is \(\frac{1}{4}\), or 25%.
By following these steps, you can tackle many probability problems with confidence!