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Chapter 5: Problem 19

Assume that random guesses are made for eight multiple choice questions on an SAT test, so that there are \(n=8\) trials, each with probability of success (correct) given by \(p=0.20 .\) Find the indicated probability for the number of correct answers. Find the probability of no correct answers.

Short Answer

Expert verified
The probability of no correct answers is approximately 0.1678.

Step by step solution

01

Identify the Type of Distribution

This is a binomial distribution problem because it involves a fixed number of trials (8 multiple-choice questions) and each trial has two possible outcomes: correct or incorrect.
02

Set Up the Binomial Probability Formula

The binomial probability formula is used here: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n = 8 \), \( k = 0 \), and \( p = 0.20 \).
03

Calculate the Binomial Coefficient

The binomial coefficient \( \binom{8}{0} \) is calculated as: \[ \binom{8}{0} = 1 \]
04

Substitute Values into the Formula

Substitute the values \( n = 8 \), \( k = 0 \), and \( p = 0.20 \) into the binomial probability formula: \[ P(X = 0) = 1 \times (0.20)^{0} \times (0.80)^{8} \]
05

Simplify the Expression

Simplify the expression: \[ P(X = 0) = 1 \times 1 \times (0.80)^8 = (0.80)^8 \] Calculate \( (0.80)^8 \): \[ (0.80)^8 \ 鈮� 0.1678 \]

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Key Concepts

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

Probability
Probability helps us determine the likelihood of an event occurring. In our example, we want to find the chance of getting no correct answers on a multiple-choice test.

It's important to remember:
  • Probabilities range from 0 (impossible event) to 1 (certain event).
  • The probability of an event plus the probability of its complement (opposite event) equals 1.

So here, we're looking for the probability of correctly guessing zero answers out of eight questions.
Binomial Coefficient
The binomial coefficient is a key part of calculating probabilities in binomial distribution scenarios. It tells us how many ways we can choose 'k' successes out of 'n' trials. In mathematical terms, it's written as \( \binom{n}{k} \).

For our example:
  • 'n' is 8 (total questions)
  • 'k' is 0 (questions we want to get right)

The binomial coefficient \( \binom{8}{0} \) equals 1, because there's only one way to get zero correct answers: by getting all of them wrong!
Multiple Choice Questions
In this scenario, we have a multiple-choice test with eight questions. Each question has one correct answer and multiple wrong ones. We assume random guessing, and the probability of guessing correctly is given as 0.20.

This is a common setup for binomial distribution problems because:
  • There are a fixed number of trials (8 questions).
  • Each trial is independent, meaning the outcome of one question doesn't affect another.
  • Each question has only two possible outcomes: correct or incorrect.

Remember these points next time you encounter multiple-choice problems!
Statistics Calculation
Lastly, we calculate the desired probability by plugging our numbers into the binomial probability formula:
\ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \
Substituting in our values \(n = 8\), \(k = 0\), and \(p = 0.20\):
  • First, calculate the binomial coefficient: \( \binom{8}{0} = 1 \)
  • Second, plug the values in: \( P(X = 0) = 1 \times (0.20)^0 \times (0.80)^8 \)
  • Simplify: \( P(X = 0) = 1 \times 1 \times (0.80)^8 = (0.80)^8 \)
  • Finally, calculate \( (0.80)^8 \): \( (0.80)^8 \approx 0.1678 \)

So, there's about a 16.78% chance of getting no correct answers. Breaking down each step makes understanding easier!

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