91Ó°ÊÓ

Answering homework questions Each question of an online homework consists of two parts. The probability that you answer the first part of a given question correctly is 0.75 . Given that you answered the first part correctly, the probability you answer the second part correctly is \(0.60 .\) Given that your missed the first part, the probability that you answer the second part correctly is 0.40 . a. What is the probability that you answer both parts of a given question correctly? b. Find the probability that you answer one of the two parts correctly (i) using the multiplicative rule with the two possible ways you can do this and (ii) by defining this as the complement of answering correctly neither or both of the two parts. c. Are the results of the two parts independent? Explain.

Step by step solution

01

Understand the Given Probabilities

The exercise provides the following probabilities: 1. Probability of answering the first part correctly ( A_1 ) is P(A_1) = 0.75 . 2. Probability of answering the second part correctly given the first part was answered correctly ( A_2 | A_1 ) is P(A_2 | A_1) = 0.60 . 3. Probability of answering the second part correctly given the first part was missed ( A_2 | A_1' ) is P(A_2 | A_1') = 0.40 .

02

Calculate Probability of Correctly Answering Both Parts

Use the multiplication rule to find the probability of answering both parts of the question correctly (both A_1 and A_2 correct). The formula is:\[P(A_1 \text{ and } A_2) = P(A_1) \times P(A_2 | A_1)\]Substitute the given probabilities:\[P(A_1 \text{ and } A_2) = 0.75 \times 0.60 = 0.45\]

03

Calculate Probability of Answering One Part Correctly (Method 1)

Calculate the probability of getting at least one part correct using the two possible ways: (i) First part is correct and the second is incorrect, or (ii) First part is incorrect and the second is correct:\[P(A_1 \text{ and } A_2') = P(A_1) \times (1 - P(A_2 | A_1)) = 0.75 \times 0.40 = 0.30\]\[P(A_1' \text{ and } A_2) = (1 - P(A_1)) \times P(A_2 | A_1') = 0.25 \times 0.40 = 0.10\]Add the probabilities:\[P(\text{One Part Correct}) = 0.30 + 0.10 = 0.40\]

04

Calculate Probability of Answering One Part Correctly (Method 2)

Define as complement of answering neither or both parts correctly:\[P(\text{Neither or Both Correct}) = P(A_1' \text{ and } A_2') + P(A_1 \text{ and } A_2)\]\[P(A_1' \text{ and } A_2') = (1 - P(A_1)) \times (1 - P(A_2 | A_1')) = 0.25 \times 0.60 = 0.15\]\[P(\text{Neither or Both Correct}) = 0.15 + 0.45 = 0.60\]The probability of answering one part correctly is:\[P(\text{One Part Correct}) = 1 - 0.60 = 0.40\]

05

Check for Independence of Events

Two events are independent if P(A_2) = P(A_2 | A_1) and P(A_2) = P(A_2 | A_1').From the given probabilities, these are not equal:\[P(A_2 | A_1) = 0.60 eq P(A_2 | A_1') = 0.40\]This discrepancy indicates that the results of the two parts are not independent.

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Key Concepts

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

Multiplication Rule

The multiplication rule is a fundamental concept in probability that helps us determine the likelihood of two events happening together. It's particularly useful in situations where the occurrence of one event affects the probability of another event. In the context of our exercise, we're dealing with two parts of a homework question. To find the probability that both parts are answered correctly, we use the multiplication rule.
The rule itself is straightforward:

• The probability of both events occurring (let's call them Event A and Event B) is the probability of Event A multiplied by the probability of Event B happening given that Event A has occurred.

For our exercise:

• Event A is answering the first part correctly, with a probability of 0.75.
• Event B is answering the second part correctly, given the first part was answered correctly, with a probability of 0.60.

Thus, the multiplication rule comes into play as:\[ P(A \text{ and } B) = P(A) \times P(B | A) = 0.75 \times 0.60 = 0.45 \]This tells us that there's a 45% chance of answering both parts correctly, showcasing how the multiplication rule is effectively applied in probability theory in real-world scenarios.

Probability Theory

Probability theory forms the backbone of understanding how likely events are to occur. It uses mathematical principles to calculate the chances or likelihood of different outcomes. In our scenario, probability allows us to quantify our chances of success in answering the homework questions.
Here are some fundamental concepts within probability theory related to our exercise:

• Sample Space: This is the set of all possible outcomes. Our sample space includes different scenarios of answering (or not) both parts of the question.
• Event: An event is a specific outcome or set of outcomes we are interested in, such as answering at least one part correctly.
• Complementary Events: These involve the concept where the probability of an event not occurring is 1 minus the probability of it occurring. This was used in the exercise to calculate the probability of answering one part correctly by considering the complement of neither being correct.

Understanding these basic components helps us apply probability theory to a wide range of real-life questions, similar to gauging our chances with different parts of homework problems.

Independence

In probability, two events are said to be independent if the occurrence of one does not affect the likelihood of the other occurring. For our exercise, we are challenged to verify if the results of answering the two parts of a homework question are independent.
The concept of independence is checked by comparing probabilities:

• If Event A and Event B are independent, then the probability of Event B should be the same regardless of whether Event A occurred or not.
• Mathematically, this means \( P(B) = P(B | A) \) and \( P(B) = P(B | A') \).

In our example, the second part being answered correctly is contingent upon the outcome of the first part:

• \( P(A_2 | A_1) = 0.60 \)
• \( P(A_2 | A_1') = 0.40 \)

Since these probabilities are different, the events are not independent. This means that the result of one part does indeed influence the likelihood of success in the other, emphasizing a key point that understanding event dependence is crucial in probability theory.