4. Probability
Multiplication Rule: Independent Events
1:59 minutesProblem 3.2.30b
Textbook Question
Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
30. Standardized Test Scores According to a survey, 57.8% of college-seeking high school seniors say they have taken one of the standardized tests for potential college students. Of these, 35.6% say they do not plan to submit their score with their college applications. (Adapted from Niche)
b. Find the probability that a randomly selected college-seeking high school senior took one of the standardized tests and plans to submit this score with their college
applications.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the probability that a randomly selected college-seeking high school senior took one of the standardized tests AND plans to submit this score with their college applications. This involves using the Multiplication Rule for probabilities.
Step 2: Recall the Multiplication Rule. The rule states that the probability of two events A and B occurring together (denoted as P(A and B)) is given by P(A and B) = P(A) * P(B|A), where P(B|A) is the probability of event B occurring given that event A has occurred.
Step 3: Define the events. Let event A be 'a student took one of the standardized tests' and event B be 'a student plans to submit their score with their college applications.' From the problem, P(A) = 57.8% = 0.578, and P(B|A) = 1 - 35.6% = 64.4% = 0.644 (since 35.6% do NOT plan to submit their scores).
Step 4: Apply the Multiplication Rule. Substitute the given probabilities into the formula: P(A and B) = P(A) * P(B|A). This becomes P(A and B) = 0.578 * 0.644.
Step 5: Interpret the result. The product from Step 4 will give the probability that a randomly selected college-seeking high school senior took one of the standardized tests AND plans to submit their score with their college applications. Perform the multiplication to find the final probability.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Multiplication Rule
The Multiplication Rule in probability states that the probability of two independent events occurring together is the product of their individual probabilities. This rule is essential for calculating the likelihood of combined events, especially when determining the probability of one event happening given that another event has already occurred.
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Conditional Probability
Conditional probability refers to the probability of an event occurring given that another event has already occurred. In this context, it helps in understanding the likelihood that a student who has taken a standardized test also plans to submit their scores, which is crucial for applying the Multiplication Rule effectively.
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Joint Probability
Joint probability is the probability of two events happening at the same time. In this scenario, it involves calculating the probability that a college-seeking high school senior both took a standardized test and intends to submit their scores. This concept is vital for combining the probabilities derived from the Multiplication Rule and conditional probabilities.
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