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Chapter 9: Problem 15

If a single die is rolled, what is the probability that it will come up as a number less than \(3 ?\)

Short Answer

Expert verified
The probability is \( \frac{1}{3} \).

Step by step solution

01

- Identify the total possible outcomes

A standard die has six faces. This means there are 6 possible outcomes when a single die is rolled: \(1, 2, 3, 4, 5, 6\).
02

- Identify the favorable outcomes

The problem asks for the probability that the die will come up as a number less than \(3 ?\). The numbers less than 3 on a die are 1 and 2. These are our favorable outcomes.
03

- Calculate the probability

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. The number of favorable outcomes is 2 (for 1 and 2), and the total number of possible outcomes is 6. Therefore, the probability is \(\frac{2}{6} = \frac{1}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

single die
A die is a small cube with numbers ranging from 1 to 6, each represented on one of its six faces. When you roll a single die, it's like spinning a six-sided wheel where each number has an equal chance of landing face up. This property ensures that each outcome (1 through 6) is equally likely, which is a key concept in probability theory. We rely on this fairness when calculating probabilities in dice-based scenarios.
favorable outcomes
To solve probability questions, it's crucial to identify 'favorable outcomes'. These are the specific results that meet the criteria of the question. In our exercise, we are asked to find the probability that a number less than 3 comes up when rolling a single die. The numbers less than 3 are 1 and 2, so these are our favorable outcomes. Understanding favorable outcomes is essential because they determine the numerator in our probability calculation.
total possible outcomes
The total possible outcomes are all the distinct results that can occur in an experiment. For a single standard die, this means all the numbers that could potentially be rolled: 1, 2, 3, 4, 5, and 6. That's 6 possible outcomes in total. Knowing the total possible outcomes is fundamental, as it sets the stage for calculating probabilities by serving as the denominator in probability formulas.
basic probability
Basic probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. In mathematical terms, if an event has 'favorable outcomes' and there are 'total possible outcomes' in all, the probability (P) of the event is given by the formula: \[P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}}\]. For our die-rolling example, the calculation becomes \[P = \frac{2}{6} = \frac{1}{3}\]. This means there is a one-in-three chance of rolling a number less than 3.

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Most popular questions from this chapter

Refer to the table. The table gives the average ages (in years) for U.S. women and men married for the first time for selected years. $$\begin{array}{l|l|c} & \text { Men } & \text { Women } \\ \hline 1940 & 24.3 & 21.5 \\ \hline 1960 & 22.8 & 20.3 \\ \hline 1980 & 24.7 & 22.0 \\ \hline 2000 & 26.8 & 25.1 \\ \hline \end{array}$$ By how much has the average age for men increased between 1940 and \(2000 ?\)

The data shown here give the average height for girls based on age. a. Make a line graph to illustrate these data. That is, write the table entries as ordered pairs and graph the points. b. Use the line graph from part (a) to predict the average height of a 10 -year-old girl. (Answers may vary.) $$\begin{array}{c|c} \hline \text { Age, } x & \text { Height (in.), } y \\ \hline 2 & 35 \\ \hline 3 & 38.5 \\ \hline 4 & 41.5 \\ \hline 5 & 44 \\ \hline 6 & 46 \\ \hline 7 & 48 \\ \hline 8 & 50.5 \\ \hline 9 & 53 \\ \hline \end{array}$$ (Bar graph can't copy)

Use a protractor to construct an angle of the given measure. $$125^{\circ}$$

Refer to the table. The table gives the average ages (in years) for U.S. women and men married for the first time for selected years. $$\begin{array}{l|l|c} & \text { Men } & \text { Women } \\ \hline 1940 & 24.3 & 21.5 \\ \hline 1960 & 22.8 & 20.3 \\ \hline 1980 & 24.7 & 22.0 \\ \hline 2000 & 26.8 & 25.1 \\ \hline \end{array}$$ What is the difference between the men's and women's average age at first marriage in \(1940 ?\)

Use a protractor to construct an angle of the given measure. $$195^{\circ}$$
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