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Chapter 10: Problem 50

In Exercises \(49-52,\) a single die is rolled twice. Find the probability of rolling a 5 the first time and a 1 the second time.

Short Answer

Expert verified
The probability of rolling a 5 on the first roll and a 1 on the second roll of a fair six-sided die is \( \frac{1}{36}. \)

Step by step solution

01

Understand the Problem

The problem is asking for the probability of rolling a 5 on the first roll and a 1 on the second roll of a fair six-sided die. One key bit of information is that the rolls are independent events, meaning that the outcome of the first roll doesn't influence the outcome of the second roll.
02

Determine the Total Number of Outcomes

For a single roll of a six-sided die, there are six possible outcomes (1, 2, 3, 4, 5, 6). Since we're rolling the die twice, we have to multiply the number of outcomes for each roll to get the total number of outcomes. So the total number of outcomes is \(6 \times 6 = 36.\)
03

Determine the Favorable Outcomes

In this case, a favorable outcome is defined as rolling a 5 on the first roll and a 1 on the second roll. Since these are independent events and there's only one way to roll a 5 and one way to roll a 1, there's only one combination of dice rolls that results in this outcome.
04

Calculate the Probability

The probability of an event is calculated by taking the number of favorable outcomes and dividing it by the total number of outcomes. So, the probability of rolling a 5 on the first roll and a 1 on the second roll is \( \frac{1}{36}.\)

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Key Concepts

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

Understanding Independent Events
When we talk about independent events in probability, we mean that the outcome of one event does not affect the outcome of the other. In the context of dice rolling, each roll of the die is an independent event. This means:

  • Rolling a 5 on the first roll has no influence on what number will appear on the second roll.
  • The likelihood of rolling any number on a die remains constant with each roll.
Understanding this concept is crucial because it allows us to multiply probabilities of separate events to find the combined probability. If two events, A and B, are independent, then the probability of both events happening is simply the product of their individual probabilities, or mathematically: \[ P(A \text{ and } B) = P(A) \times P(B) \]For our exercise, since both rolls are independent, the calculation is straightforward.
Exploring Dice Probabilities
Dice probabilities are a great way to introduce fundamental probability concepts. A standard six-sided die has six faces, numbered from 1 to 6. This gives six equally likely outcomes. When considering dice probabilities:
  • Each side has a probability of \( \frac{1}{6} \). This is because there's one way to roll each number and six possible outcomes in total.
  • To find the total number of outcomes for multiple rolls, multiply the outcomes of each roll.
For instance, if you roll a die twice, the outcomes for both rolls together are \( 6 \times 6 = 36 \). This means there are 36 possible results when rolling a die twice. This concept allows us to understand how individual roll results influence combined outcomes.
Calculating Outcomes in Probability
The concept of outcomes in probability involves identifying all possible results of a random experiment and distinguishing which are favorable. For our dice rolling exercise:
  • The total possible outcomes from rolling a die twice are 36.
  • A favorable outcome in this context is the specific event of rolling a 5 first and then a 1.
  • This particular combination occurs once in 36 possible outcomes.
To calculate the probability of rolling a 5 followed by a 1, use the formula:
\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]
Thus, the probability is \( \frac{1}{36} \), illustrating how understanding outcomes can quantify the likelihood of specific events.

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Most popular questions from this chapter

Make Sense? In Exercises \(66-69\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Assuming the next U.S. president will be a Democrat or a Republican, the probability of a Republican president is 0.5

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