5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
1:59 minutesProblem 5.C.4a
Textbook Question
Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (a) fewer than four personal fouls,
Verified step by step guidance1
Identify the probability distribution provided in Exercise 3. This could be a discrete probability distribution, such as a probability mass function (PMF), or a table listing the probabilities for each possible number of personal fouls.
Determine the range of values that correspond to 'fewer than four personal fouls.' This means you are interested in the probabilities for 0, 1, 2, and 3 personal fouls.
Sum the probabilities for the values identified in the previous step. Mathematically, this can be expressed as: P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).
If the probabilities are given in a table, locate the corresponding values for P(X = 0), P(X = 1), P(X = 2), and P(X = 3) and add them together.
Verify that the total probability for all possible outcomes in the distribution equals 1. This ensures the distribution is valid and that 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, allowing us to understand the behavior of the variable. In this context, it helps in determining the probability of DeMar DeRozan having fewer than four personal fouls based on the distribution of his fouls in previous games.
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Random Variable
A random variable is a numerical outcome of a random phenomenon, which can take on different values based on chance. In this case, the random variable is the number of personal fouls committed by DeMar DeRozan in a game. Understanding random variables is crucial for calculating probabilities and making inferences about the data.
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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. It is essential for answering the question about the likelihood of DeRozan having fewer than four personal fouls, as it involves summing the probabilities of all outcomes that meet this criterion. This concept allows us to assess the overall likelihood of a range of outcomes.
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