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Chapter 4: Problem 14

Express the indicated degree of likelihood as a probability value between \(\boldsymbol{0}\) and \(\boldsymbol{I}\).When making a random guess for an answer to a multiple choice question on an SAT test, the possible answers are \(a, b, c, d, e,\) so there is 1 chance in 5 of being correct.

Short Answer

Expert verified
The probability is 0.2.

Step by step solution

01

Identify Total Possible Outcomes

Determine the total number of possible answers for the multiple choice question. In this case, the possible answers are \(a, b, c, d, e\). Hence, the total number of possible outcomes is 5.
02

Identify the Number of Favorable Outcomes

Identify the number of outcomes that would be considered successful or correct. In this scenario, there is only 1 correct answer out of the 5 possible answers.
03

Calculate the Probability

Use the probability formula to calculate the likelihood of choosing the correct answer. The probability formula is given by \[ P(\text{event}) = \frac{ \text{Number of Favorable Outcomes} }{ \text{Total Number of Possible Outcomes} } \]. Substituting the values, the probability is \[ P(\text{correct answer}) = \frac{1}{5} \].
04

Express the Probability as a Decimal

Convert the fraction to a decimal value. \[ \frac{1}{5} = 0.2 \]. Therefore, the probability of guessing the correct answer is 0.2.

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Key Concepts

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

Random Guessing
Random guessing implies choosing an answer without any prior knowledge or strategy. Often, in exams like the SAT, this means picking one of the multiple choice answers purely by chance.
When you don't know the answer to a question, and you guess randomly, every option has an equal probability of being chosen.

Let's illustrate this with a multiple choice question that has 5 possible answers: a, b, c, d, and e. Since you are guessing, the chance you pick the correct answer is determined by the number of answers.
  • In this example, there are 5 possible answers.
  • Only one of these answers is correct.
Therefore, the probability of randomly guessing the correct answer is low, specifically 1 out of 5 or 20%.
Multiple Choice Questions
Multiple choice questions are a common type of test question where you select the correct answer from several options. In our example of a multiple choice question with answers: a, b, c, d, e, the structure is clear:

Understanding how the choices work is crucial for calculating probabilities:
  • The total number of potential choices is 5.
  • Out of these choices, only one is correct.
When considering how to improve the chance of guessing correctly, remember that each incorrect answer reduces the chance of a correct guess. This is why preparation and strategy are better approaches than guessing.
However, knowing the basic probability helps in understanding the risk involved with random guessing and informs better decision-making when faced with uncertainty.
Fraction to Decimal Conversion
Converting fractions to decimals is a fundamental skill in understanding probabilities. Often, probabilities are expressed as fractions but need to be interpreted as decimals for a clearer understanding.
The probability of guessing the correct answer in our example is \(\frac{1}{5}\). Here's how to convert it:
  • Recognize that the fraction represents the division of 1 by 5.
  • Perform the division: 1 divided by 5 equals 0.2.
So, the fraction \(\frac{1}{5}\) converts to the decimal 0.2. This means that there is a 20% chance of guessing correctly.
Understanding this conversion is crucial not only for classroom problems but also for interpreting statistics and odds in real life.
You'll often move between fractions and decimals, so mastering this skill will greatly help in various mathematical contexts.

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Most popular questions from this chapter

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