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Chapter 5: Problem 74

At the gym Suppose that \(10 \%\) of adults belong to health clubs, and \(40 \%\) of these health club members go to the club at least twice a week. What percent of all adults go to a health club at least twice a week? Write the information given in terms of probabilities, and use the general multiplication rule.

Short Answer

Expert verified
4% of adults go to a health club at least twice a week.

Step by step solution

01

Identify the Given Information

We are given that 10% of adults belong to health clubs, so the probability that a randomly chosen adult belongs to a health club is \( P(A) = 0.10 \). Furthermore, 40% of these health club members go to the club at least twice a week. Therefore, the probability that a randomly chosen health club member goes at least twice a week is \( P(B|A) = 0.40 \).
02

Use the General Multiplication Rule

The general multiplication rule of probability states that \( P(A \cap B) = P(A) \cdot P(B|A) \). Here, \( A \cap B \) represents the event of an adult both belonging to a health club and attending it at least twice a week.
03

Calculate \( P(A \cap B) \)

Substitute the given probabilities into the formula: \[ P(A \cap B) = 0.10 \times 0.40 = 0.04 \].
04

Interpret the Result

The result \( P(A \cap B) = 0.04 \) means that 4% of all adults both belong to a health club and attend it at least twice a week.

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Key Concepts

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

General Multiplication Rule
The General Multiplication Rule is a cornerstone in the study of probability. It helps you find the likelihood of two events both occurring. Imagine you have two events, A and B. The rule gives us the probability of both A and B happening together. It is expressed mathematically as \( P(A \cap B) = P(A) \times P(B|A) \).
Here鈥檚 a breakdown:
  • \( P(A) \) is the probability of the first event, A, happening.
  • \( P(B|A) \) is the conditional probability of event B happening given that A has already occurred.
In practical terms, if you know how likely first an event is, and how likely a second event is given the first event happened, you can determine how often you would expect both events to occur together. Thus, applying this in our exercise, we use it to calculate how many adults both belong to a health club and go at least twice a week.
Conditional Probability
Conditional Probability is when you want to know how likely something is to happen, given that something else has already occurred. In symbols, it鈥檚 represented as \( P(B|A) \). This means the probability of event B occurring, given event A is true.
It鈥檚 extremely useful because it narrows down an otherwise broad event into more specific scenarios.
  • For example, in our gym scenario, 40% of those who are already health club members go to the gym at least twice a week.
This concept hones our focus to only those who meet a certain condition (being a health club member in this context). When we understand conditional probability, it helps us interpret data and make informed predictions based on existing information. In essence, conditional probability gives us a more precise picture.
Statistics Education
Statistics Education is all about helping people grasp concepts of data analysis, probabilities, and interpreting statistical information effectively. Students learn about different statistical tools and how to apply them in real-world scenarios.
Understanding probability rules, like the general multiplication rule and conditional probability, is essential in Statistics Education. It prepares students to handle data analytically. Here鈥檚 why it鈥檚 crucial:
  • It equips learners with the skills to predict outcomes and make decisions based on statistical data.
  • In the gym scenario, it helps students see how interconnected probabilities are used to solve real-life problems.
  • This form of education fosters critical thinking and enhances decision-making.
Statistics Education empowers learners to translate mathematical concepts into practical solutions. Like the gym membership problem, it鈥檚 not just about numbers; it鈥檚 about what those numbers mean in our everyday lives.

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Most popular questions from this chapter

Mac or \(\mathrm{PC}\) ? A recent census at a major university revealed that \(40 \%\) of its students mainly used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, \(67 \%\) of the school's students were undergraduates. The rest were graduate students. In the census, \(23 \%\) of respondents were graduate students who said that they used \(\mathrm{PCs}\) as their main computers. Suppose we select a student at random from among those who were part of the census. (a) Make a two-way table for this chance process. (b) Construct a Venn diagram to represent this setting. (c) Consider the event that the randomly selected student is a graduate student and uses a Mac. Write this event in symbolic form based on your Venn diagram in part (b). (d) Find the probability of the event described in part (c). Explain your method.

Mammograms Many women choose to have annual mammograms to screen for breast cancer after age 40\. A mammogram isn't foolproof. Sometimes the test suggests that a woman has breast cancer when she really doesn't (a "false positive"). Other times the test says that a woman doesn't have breast cancer when she actually does (a "false negative"). Suppose the false negative rate for a mammogram is \(0.10 .\) (a) Interpret this probability as a long-nun relative frequency. (b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

A basketball player claims to make \(47 \%\) of her shots from the field. We want to simulate the player taking sets of 10 shots, assuming that her claim is true. To simulate the number of makes in 10 shot attempts, you would perform the simulation as follows: (a) Use 10 random one-digit numbers, where \(0-4\) are a make and \(5-9\) are a miss. (b) Use 10 random two-digit numbers, where \(00-46\) are a make and \(47-99\) are a miss. (c) Use 10 random two-digit numbers, where \(00-47\) are a make and \(48-99\) are a miss. (d) Use 47 random one-digit numbers, where 0 is a make and \(1-9\) are a miss. (e) Use 47 random two-digit numbers, where \(00-46\) are a make and \(47-99\) are a miss.

Free throws A basketball player has probability 0.75 of making a free throw. Explain how you would use each chance device to simulate one free throw by the player. (a) A standard deck of playing cards (b) Table D of random digits (c) A calculator or computer's random integer generator

Roulette An American roulette wheel has 38 slots with numbers 1 through \(36,0,\) and \(00,\) as shown in the figure. Of the numbered slots, 18 are red, 18 are black, and 2 - the 0 and 00 - are green. When the wheel is spun, a metal ball is dropped onto the middle of the wheel. If the wheel is balanced, the ball is equally likely to settle in any of the numbered slots. Imagine spinning a fair wheel once. Define events \(B\) : ball lands in a black slot, and \(E\) : ball lands in an even-numbered slot. (Treat 0 and 00 as even numbers. \()\) (a) Make a two-way table that displays the sample space in terms of events \(B\) and \(E\). (b) Find \(P(B)\) and \(P(E)\). (c) Describe the event \({ }^{\alpha} B\) and \(E^{\prime \prime}\) in words. Then find \(P(B\) and \(E)\) (d) Explain why \(P(B\) or \(E) \neq P(B)+P(E) .\) Then use the general addition rule to compute \(P(B\) or \(\bar{E})\)
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