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Chapter 26: Problem 24

The probability of getting a total of 5 when a pair of dice is torieed is \(\ldots\)

Short Answer

Expert verified
The probability is \( \frac{1}{9} \).

Step by step solution

01

Understand the Problem

When a pair of dice is rolled, we want to find the probability of the numbers on the two dice adding up to exactly 5.
02

List All Possible Outcomes

Each die has 6 sides, so there are a total of 6 脳 6 = 36 possible outcomes when rolling two dice.
03

Identify Favorable Outcomes

List the dice combinations that add up to 5: (1,4), (2,3), (3,2), and (4,1). There are 4 favorable outcomes.
04

Calculate the Probability

Probability is the ratio of favorable outcomes to the total number of outcomes. Therefore, the probability of rolling a total of 5 is \( \frac{4}{36} = \frac{1}{9} \).

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Key Concepts

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

Dice Probability
When discussing dice probability, we look at the outcome of rolling one or more dice. Dice are fair, six-sided objects commonly used in probability exercises. Each face of a die shows a different number from 1 to 6.

In the case of two dice being rolled, each die operates independently. This means the result of the first die doesn't affect that of the second. The total number of outcomes for two dice can be calculated by multiplying the number of outcomes for each die.
  • One die has 6 possible outcomes.
  • Two dice together have 6 脳 6 = 36 possible outcomes.
Whether you're a beginner or seasoned mathematician, understanding that each roll of a die has an equal probability is crucial in probability theory.
Favorable Outcomes
Favorable outcomes are those specific outcomes that meet the conditions of the problem at hand. In probability exercises involving dice, identifying these is key. For example, if you're asked to get a certain sum, you need to find all dice combinations that result in that sum.

In our example of rolling a total of 5:
  • The dice pair combinations are: (1,4), (2,3), (3,2), and (4,1).
  • Thus, there are 4 favorable outcomes as these combinations add up to 5.
Listing and counting these favorable outcomes helps to clarify the structure of the problem and plays a crucial role in calculating probability.
Calculating Probability
Calculating probability involves determining how likely an event is to occur. The probability of an event is found by dividing the number of favorable outcomes by the total possible outcomes.

Let's break it down using our dice example:
  • The total number of outcomes from rolling two dice is 36.
  • The number of outcomes where the sum is 5 is 4.
  • To find the probability, divide the number of favorable outcomes by the total number of outcomes: \( \frac{4}{36} \).
  • This simplifies to \( \frac{1}{9} \).
This fraction represents the chance of rolling a total of 5 with two dice. Calculating probability gives us a numeric sense of how likely an event will happen, which is a cornerstone of probability theory.

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