4. Probability
Addition Rule
3:36 minutesProblem 4.2.33
Textbook Question
Exclusive Or The exclusive or means either one or the other event occurs, but not both.
If one of the high school drivers is randomly selected, find the probability of getting one who texted while driving or drove when drinking alcohol.
b. Repeat Exercise 11 “Texting or Drinking†using the exclusive or instead of the inclusive or.
Verified step by step guidance1
Understand the concept of 'exclusive or': The exclusive or (XOR) means that either one event occurs or the other event occurs, but not both. This is different from the inclusive or, which allows for both events to occur simultaneously.
Identify the probabilities of the individual events: Let P(A) represent the probability of texting while driving, and P(B) represent the probability of driving when drinking alcohol. These probabilities should be provided in the problem or calculated from the data.
Identify the probability of both events occurring simultaneously: Let P(A ∩ B) represent the probability that a driver both texted while driving and drove when drinking alcohol. This value should also be provided or calculated from the data.
Apply the formula for the probability of the exclusive or: The formula for the exclusive or is P(A XOR B) = P(A) + P(B) - 2 * P(A ∩ B). This formula ensures that the overlap (both events occurring) is subtracted twice to exclude it entirely.
Substitute the known values into the formula: Replace P(A), P(B), and P(A ∩ B) with their respective values to calculate the probability of a driver who either texted while driving or drove when drinking alcohol, but not both.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exclusive Or (XOR)
The exclusive or (XOR) is a logical operation that results in true if and only if one of the operands is true, but not both. In probability, this means that when calculating the likelihood of two events, we consider only the scenarios where one event occurs without the other. For example, if we are looking at the events of texting while driving and drinking while driving, XOR would mean we are interested in cases where a driver either texted or drank, but not both.
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Probability of Mutually Exclusive Events
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of the question, we are interested in calculating the probability of selecting a high school driver who either texted while driving or drove after drinking alcohol. This involves determining the number of favorable outcomes over the total number of possible outcomes.
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Mutually Exclusive Events
Mutually exclusive events are events that cannot occur at the same time. In the context of the exclusive or, if one event occurs, the other cannot. For instance, if a driver is texting while driving, they cannot simultaneously be drinking alcohol. Understanding this concept is crucial for correctly applying the XOR in probability calculations, as it simplifies the process of determining the total probability of either event occurring.
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