A 50-question true-false exam is given. Each correct answer is worth 10 points. Consider an unprepared student who randomly guesses on each question. In your solution, please be sure to show the exact steps you used to arrive at the answers.
A. If 5 points are deducted for each incorrect answer, what is the chance that the student will score at least 200 points?
B. If 10 points are deducted for each wrong answer, what is the chance that the student will get a negative score on the exam?
The closing price of Schnur Sporting Goods, Inc., common stock is uniformly distributed between $15 and 30 per share.
What is the probability that the stock price will be:
(a) More than $24? (Round your answer to 4 decimal places.)
Probability
(b) Less than or equal to $17? (Round your answer to 4 decimal places.)
Probability
To solve these questions, we need to use the concept of probability. Let's break them down step by step:
A. If 5 points are deducted for each incorrect answer, what is the chance that the student will score at least 200 points?
1. First, we need to find the highest possible score the student can achieve. With 50 true-false questions, and each correct answer worth 10 points, the maximum score is 50 * 10 = 500 points.
2. Now, let's calculate the minimum number of correct answers the student needs to score at least 200 points. Let's call this value x. We can find it using the formula: x * 10 - (50 - x) * 5 = 200.
Simplifying this equation, we get: 10x - (50 - x) * 5 = 200.
Expanding the bracket, we get: 10x - 250 + 5x = 200.
Combining like terms, we get: 15x - 250 = 200.
Adding 250 to both sides: 15x = 450.
Dividing both sides by 15: x = 30.
Therefore, the student needs to answer at least 30 questions correctly to score at least 200 points.
3. Now, let's find the probability of the student answering at least 30 questions correctly by guessing randomly on each question. For a true-false question, the probability of guessing correctly is 1/2.
The probability of getting exactly x questions correct when guessing randomly is given by the binomial probability formula: P(x) = nCx * p^x * (1-p)^(n-x).
In this case, n (the number of trials) is 50, x is at least 30, p (the probability of success) is 1/2, and (1-p) is also 1/2 since we have only two options (true or false).
Therefore, the probability of scoring at least 200 points can be found by adding up the individual probabilities for each value of x from 30 to 50 (inclusive).
B. If 10 points are deducted for each wrong answer, what is the chance that the student will get a negative score on the exam?
1. Similar to the previous question, we need to find the maximum possible score the student can achieve. With 50 true-false questions, and each correct answer being worth 10 points, the maximum score is 50 * 10 = 500 points.
2. Now, let's calculate the maximum number of incorrect answers the student can afford to score a negative overall score. Let's call this value y.
We can find it using the formula: 500 - y * 10 - (50 - y) * 10 < 0.
Simplifying this equation, we get: 500 - 10y - 500 + 10y < 0.
Canceling out terms, we get: -10y < 0.
Dividing by -10 (and flipping the inequality sign), we get: y > 0.
Therefore, the student can score any value of incorrect answers greater than 0 without getting a negative overall score.
3. Since there are an infinite number of incorrect answers greater than 0 that the student can score, the probability of getting a negative overall score is 0.
So, to summarize:
A. The chance that the student will score at least 200 points is found by summing the individual probabilities for each value of x from 30 to 50 (inclusive), where x is the number of correct answers.
B. The chance that the student will get a negative score is 0, as there are an infinite number of incorrect answers greater than 0.