4. Probability
Basic Concepts of Probability
1:58 minutesProblem 4.q.8
Textbook Question
In Exercises 6–10, use the following results from tests of an experiment to test the effectiveness of an experimental vaccine for children (based on data from USA Today). Express all probabilities in decimal form.
If 1 of the 1602 subjects is randomly selected, find the probability of getting 1 who had the vaccine treatment and developed flu.
Verified step by step guidance1
Step 1: Identify the total number of subjects in the experiment. Add all the values in the table: 14 (Vaccine Treatment, Developed Flu) + 1056 (Vaccine Treatment, Did Not Develop Flu) + 95 (Placebo, Developed Flu) + 437 (Placebo, Did Not Develop Flu).
Step 2: Focus on the specific group mentioned in the problem: subjects who had the vaccine treatment and developed flu. From the table, this corresponds to the value 14.
Step 3: Calculate the probability by dividing the number of subjects in the specified group (14) by the total number of subjects (calculated in Step 1). Use the formula: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).
Step 4: Express the probability in decimal form. Ensure the division result is converted to a decimal.
Step 5: Verify the calculation by double-checking the values from the table and ensuring the probability is correctly expressed in decimal form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it refers to the chance of randomly selecting a subject who received the vaccine treatment and developed flu. The probability can be calculated by dividing the number of favorable outcomes by the total number of outcomes.
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Favorable Outcomes
Favorable outcomes are the specific results that we are interested in when calculating probability. In this case, the favorable outcome is the number of subjects who received the vaccine treatment and developed flu, which is 14. Understanding favorable outcomes is crucial for accurately determining the probability of an event.
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Total Outcomes
Total outcomes refer to the complete set of possible results in a probability scenario. Here, the total number of subjects is 1602, which includes both those who received the vaccine and those who received the placebo. Knowing the total outcomes is essential for calculating the probability, as it serves as the denominator in the probability formula.
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