91Ó°ÊÓ

Working out Choose a person aged 19 to 25 years at random and ask, "In the past seven days, how many times did you go to an exercise or fitness center or work out?" Call the response \(Y\) for short. Based on a large sample survey, here is a probability model for the answer you will get: \({ }^{8}\) $$ \begin{array}{lcccccccc} \hline \text { Days: } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text { Probability: } & 0.68 & 0.05 & 0.07 & 0.08 & 0.05 & 0.04 & 0.01 & 0.02 \\\ \hline \end{array} $$ (a) Show that this is a legitimate probability distribution. (b) Make a histogram of the probability distribution. Describe what you see. (c) Describe the event \(Y<7\) in words. What is \(P(Y<7) ?\) (d) Express the event "worked out at least once" in terms of \(Y\). What is the probability of this event?

Step by step solution

01

Verify Probability Distribution (a)

To verify the probability distribution is legitimate, check if all probabilities add up to 1. Add the probabilities given: 0.68 + 0.05 + 0.07 + 0.08 + 0.05 + 0.04 + 0.01 + 0.02. The sum is 1, so this is a valid probability distribution.

02

Construct Histogram (b)

Create a bar graph where the x-axis represents 'Days' (0 through 7), and the y-axis represents 'Probability'. Draw each bar with its corresponding height based on the probabilities: 0.68 for Day 0, 0.05 for Day 1, etc. The histogram will show that most individuals didn't work out (high bar at 0 days), with probabilities decreasing as daily workout frequency increases.

03

Describe Event Y

The event \(Y<7\) means that a person worked out on either 0, 1, 2, 3, 4, 5, or 6 days but not on all 7 days of the week. Calculate \(P(Y<7)\) by summing probabilities for days 0 through 6: 0.68 + 0.05 + 0.07 + 0.08 + 0.05 + 0.04 + 0.01 = 0.98.

04

Event Worked Out At Least Once (d)

The event "worked out at least once" is described by \(Y\geq 1\). The probability is the sum of probabilities for days 1 through 7. Alternatively, calculate it as 1 minus the probability of not working out at all (\(Y=0\)): \(1 - P(Y=0) = 1 - 0.68 = 0.32\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Model

A probability model is a mathematical representation that helps us understand random phenomena. It consists of two main components: a sample space and a probability distribution. The sample space includes all possible outcomes of a random experiment, while the probability distribution assigns probabilities to each of these outcomes.
For the exercise we are discussing, the sample space consists of the number of days a person aged 19 to 25 works out in a week, which ranges from 0 to 7 days. Each possible outcome (or number of workout days) has an associated probability. We have seen that these probabilities add up to 1, which is a key property of any valid probability model. This means all possible outcomes and their probabilities are accounted for and the model represents real-world behavior accurately.

Histogram

A histogram is a type of bar graph that displays the frequency of various outcomes in a dataset. In probability, it is used to visualize the probability distribution of a random variable. By doing so, it provides a clear visual summary of how the probabilities are distributed across different outcomes.
In the provided exercise, the histogram showcases the probabilities for each possible number of workout days, from 0 to 7. With the exercise data, the histogram shows a tall bar at 0 days indicating the majority of people did not work out at all, and increasingly shorter bars as the number of workout days increases. This visual representation helps us understand the skewness and concentration of the distribution easily and effectively.

Event Description

When discussing probabilities, an event refers to a set of outcomes to which a probability is assigned. Events can be simple, involving a single outcome, or more complex, incorporating multiple outcomes.
For example, the event "Y<7" in our exercise means that an individual exercised less than 7 days in the past week. This involves summing up the probabilities of working out 0 through 6 days. This approach gives a comprehensive view of how often people did not workout each specific number of days or less than every day, revealing behavioral patterns across the young adult group.

Probability Calculation

Probability calculation is the process of determining the likelihood of an event. It is vital for making informed predictions and decisions based on data. For successful probability calculation, it is necessary to have a clear event description and the probabilities of each outcome.
In the exercise context, calculating the probability of working out at least once is achieved by considering all days except zero. Alternatively, we use the complement rule: subtract the probability of the unwanted scenario (not working out at all) from 1. This math approach aids in determining the probability accurately, which turned out to be 0.32, giving insights into how likely this behavior pattern is among young individuals.