Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 12: Problem 25
Prisana guesses at all 10 true/false questions on her history test. Find each probability. \(P(\text { exactly } 6 \text { correct })\)
Short Answer
Expert verified
The probability of getting exactly 6 correct is approximately 0.21.
Step by step solution
01
Identify the Type of Experiment
This problem is a classic example of a Binomial Experiment since there are a fixed number of trials (10 questions), each with two possible outcomes (True or False), and a constant probability of success (getting the question correct).
02
Define Binomial Probability Formula
The probability of getting exactly k successes in n independent Bernoulli trials is given by the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) is the binomial coefficient, \( p \) is the probability of success on each trial, and \( n \) is the number of trials.
03
Identify Values for the Problem
For this problem, n = 10 (questions), k = 6 (correct answers), and p = 0.5 (probability of guessing correctly, as each answer has two possible outcomes).
04
Calculate Binomial Coefficient
The binomial coefficient \( \binom{n}{k} \) for this problem is:\[\binom{10}{6} = \frac{10!}{6! (10-6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210\]
05
Apply Binomial Probability Formula
Substitute the identified values into the binomial probability formula:\[P(X = 6) = 210 \cdot (0.5)^6 \cdot (0.5)^{4} = 210 \cdot (0.5)^{10}\]
06
Compute the Probability
Calculate the expression from Step 5:\[P(X = 6) = 210 \cdot \left(\frac{1}{1024}\right) = \frac{210}{1024} = 0.205078125\]
07
Round the Probability
The probability is often rounded to a more practical number, typically two decimal places. Thus, the probability of Prisana getting exactly 6 correct is approximately:\[P(X=6) \approx 0.21\]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Experiment
A binomial experiment is a statistical experiment that satisfies four key criteria:
- Fixed number of trials: In a binomial experiment, the number of trials, denoted by \( n \), is fixed in advance. For Prisana's test, this is 10 since she answers 10 questions.
- Two possible outcomes: Each trial has only two possible outcomes, such as "success" or "failure". In our case, success is a correct answer and failure is an incorrect one.
- Independent trials: The result of one trial does not affect the result of another. Assuming Prisana guesses each question without influencing the others, her answers fit this condition.
- Constant probability of success: The probability \( p \) of success (answering a question correctly) is the same for each trial. Here, it is 0.5 because it's a guess between two options.
Bernoulli Trials
The concept of Bernoulli trials is central to understanding binomial experiments.
Each individual trial within a binomial experiment is a Bernoulli trial. Let's break down the important aspects of a Bernoulli trial:
Each individual trial within a binomial experiment is a Bernoulli trial. Let's break down the important aspects of a Bernoulli trial:
- Single Trial Outcome: A Bernoulli trial involves a single trial, which results in a "success" or "failure". Here, success can be getting the answer right.
- Probability of Success: Numbered \( p \), it remains the same for each trial. For Prisana, each question guessed has a probability of 0.5 to be correct.
- Independence: Each trial is carried out independently, which means that knowing the outcome of one does not convey information about another.
Probability
In the context of a binomial experiment, probability refers to the chance of achieving a specific number of successful outcomes across all trials.
Probability seeks to answer the question: "What are the odds of this happening?" For Prisana, this involves calculating the odds of her getting exactly 6 questions out of 10 correct when she guesses blindly.
To calculate this probability, we employ the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here:
Probability seeks to answer the question: "What are the odds of this happening?" For Prisana, this involves calculating the odds of her getting exactly 6 questions out of 10 correct when she guesses blindly.
To calculate this probability, we employ the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here:
- \( n \) is the total number of trials, 10 in this case.
- \( k \) is the number of successes we're measuring for, which is 6.
- \( p \) is the success probability for each trial, set at 0.5.
Binomial Coefficient
The binomial coefficient is a crucial component of the binomial probability formula, often symbolized as \( \binom{n}{k} \).
This element of the formula answers a specific question: "How many ways can we choose \( k \) successes out of \( n \) trials?"
For Prisana's test, the calculation:\[\binom{10}{6} = \frac{10!}{6! (10-6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210\]provides the number of different ways she can get exactly 6 questions right. Here:
This element of the formula answers a specific question: "How many ways can we choose \( k \) successes out of \( n \) trials?"
For Prisana's test, the calculation:\[\binom{10}{6} = \frac{10!}{6! (10-6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210\]provides the number of different ways she can get exactly 6 questions right. Here:
- \( n! \) denotes the factorial of \( n \), which calculates the product of all positive integers up to \( n \).
- \( k! \) and \((n-k)!\) denote similar products for \( k \) and \( n-k \) respectively.
