Final answer:
The question involves calculating different types of probabilities related to parking cars and understanding different probability distributions, such as normal and exponential. There's a clarity issue with the initial probability distribution mentioned. The specific examples given require application of statistical methods like z-scores and examining cumulative distributions.
Step-by-step explanation:
Understanding Probability Distributions
When we convert information such as the number of hours parked into a probability distribution, we list the possible outcomes and assign a probability to each outcome such that all probabilities sum up to 1. However, it is crucial to note that in the question provided, there seems to be an inconsistency because the probabilities provided sum up to more than 1 (a. 0.106 + b. 0.170 + c. 0.221 + d. 0.289 = 0.786 and not 1) which is not possible in a valid probability distribution. To address the parts of the question not concerning probability distribution, here are the explanations:
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- The probability of waiting at least 8 minutes to find a parking space could be found by subtracting the cumulative distribution function value at 8 minutes from 1, depending on the distribution parameters.
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- To determine the percentage of time it takes more than a certain amount of minutes to find a parking space, we look for the value where the cumulative distribution exceeds 30% (since we want 70% to take longer).
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- If 37.5 percent of the cars parked are parked crookedly, then for every 22 cars, an expected number of them, which can be calculated as 0.375 * 22, will be parked crookedly.
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- The length of time for finding a parking space following a normal distribution means that the probability distribution is continuous and can be described using its mean and standard deviation.
Mathematical techniques such as the z-score calculation and exponential functions are used to solve problems related to normal or exponential distributions respectively.