5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
2:02 minutesProblem 5.CR.4c
Textbook Question
Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (c) between two and four personal fouls, inclusive.
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Identify the probability distribution provided in Exercise 3. This could be a table, formula, or list of probabilities associated with the number of personal fouls committed by DeMar DeRozan. Ensure you have the probabilities for all relevant outcomes (e.g., 0, 1, 2, 3, 4 fouls, etc.).
Determine the range of outcomes for which you need to calculate the probability. In this case, it is 'between two and four personal fouls, inclusive,' which means you need the probabilities for 2, 3, and 4 fouls.
Extract the probabilities corresponding to the outcomes of 2, 3, and 4 personal fouls from the probability distribution. For example, if \( P(X = 2) \), \( P(X = 3) \), and \( P(X = 4) \) are the probabilities, note these values.
Add the probabilities for the specified outcomes. Use the formula: \( P(2 \leq X \leq 4) = P(X = 2) + P(X = 3) + P(X = 4) \). This step combines the probabilities for the desired range.
Verify that the sum of probabilities in the entire distribution equals 1 (if not already done in Exercise 3). This ensures the distribution is valid and your calculations are based on accurate data.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability Distribution
A probability distribution describes how the probabilities of a random variable are distributed across its possible values. It provides a mathematical function that gives the likelihood of each outcome. In this context, it helps determine the probability of DeMar DeRozan having a specific number of personal fouls during a game.
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Random Variable
A random variable is a numerical outcome of a random phenomenon. In this case, the random variable represents the number of personal fouls committed by DeMar DeRozan in a game. Understanding random variables is crucial for calculating probabilities and interpreting the results of statistical analyses.
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Cumulative Probability
Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. To find the probability of DeRozan having between two and four personal fouls, inclusive, one would calculate the cumulative probabilities for these values and then find the difference to get the desired probability range.
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