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Chapter 13: Problem 18

An instant lottery game gives you probability \(0.02\) of winning on any one play. Plays are independent of each other. If you play three times, the probability that you win on none of your plays is about (a) \(0.98\). (b) \(0.94\) (c) \(0.000008\).

Short Answer

Expert verified
(b) 0.94

Step by step solution

01

Understanding Probability of Not Winning

The probability of not winning on a single play is the complement of winning, which means it is 1 minus the probability of winning. Given that the probability of winning is \(0.02\), the probability of not winning on a single play is \(1 - 0.02 = 0.98\).
02

Calculating Probability of Not Winning in All Three Plays

Since the plays are independent of each other, the probability of not winning on all three plays is the product of the probability of not winning each play. Therefore, the probability is \(0.98 \times 0.98 \times 0.98 = 0.98^3\).
03

Evaluating \(0.98^3\)

Compute \(0.98^3\) using a calculator: \(0.98^3 \approx 0.941192\). So, the probability of not winning any of the plays is approximately \(0.94\).
04

Matching with Given Options

From the computed value \(0.941192\), we see that it is approximately equal to option (b) \(0.94\).

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Key Concepts

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

Independent Events
In probability theory, understanding whether events are independent is crucial. Two or more events are said to be independent if the outcome of one event does not affect the outcome of the other. When you consider independent events, you can determine the joint probability simply by multiplying the probabilities of each event occurring. This simplifies complex probability calculations, as each event does not alter the chances of the others.
  • In our lottery game scenario, each play is independent. The chance of winning or not on one play doesn鈥檛 influence the outcomes of subsequent plays.
  • This independence allows us to calculate the probability of a series of events (such as not winning any games) by multiplying individual probabilities for each event.
When dealing with multiple plays in an independent game, you maintain simplicity in your calculations. Each play's probability remains unaffected by previous outcomes, providing a straightforward path to finding joint probabilities.
Complement Rule
The complement rule is a fundamental concept in probability, allowing us to compute the probability that an event does not occur. It is defined as the difference between 1 and the probability of the event occurring.
  • For example, if the probability of winning a game is \,\(0.02\,\), the probability of losing (the complement) is \,\(1 - 0.02 = 0.98\,\).
  • This rule helps in providing quick answers to questions about non-occurrence of events, especially useful when non-occurrence is more common than occurrence.
Using the complement rule can significantly ease the calculation process. Rather than calculating the probability of every possible unsuccessful scenario, knowing the probability of winning gives us the ability to find the probability of losing with just a simple subtraction. This is particularly helpful when events are more likely to not occur, making it a practical tool in probability computations.
Probability Calculation
Calculating probability, especially for multiple independent events, often involves simple arithmetic and multiplication of fractions or decimals.
  • To find the probability of not winning in several independent plays of a lottery game, you first calculate the probability of the complement, in this case not winning a single play.
  • Next, since the events are independent, the probability of not winning across multiple plays involves multiplying the complement probabilities: \,\(0.98 \times 0.98 \times 0.98 = 0.98^3\,\).
Despite its simplicity, introducing these calculations to real-life scenarios can greatly determine outcomes in games of chance, insurance evaluations, and risk assessments. In our example, the calculation demonstrates a controlled approach to assessing risk and likelihood in repeated independent attempts, achieving clarity in understanding chances across multiple attempts. A simple calculator can provide the final numerical probability, showcasing the accessibility and usability of probability calculations in everyday tasks.

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Most popular questions from this chapter

Independent? In 2015 , the Report on the LC Berkeley Faculty Salary Equity Study shows that 87 of the university's 222 assistant professors were women, along with 137 of the 324 associate professors and 243 of the 972 full professors. (a) What is the probability that a randomly chosen Berkeley professor (of any rank) is a woman? (b) What is the conditional probability that a randomly chosen professor is a woman, given that the person chosen is a full professor? (c) Are the rank and sex of Berkeley professors independent? How do you know?

Playing the lottery. New York State's "Quick Draw" lottery moves right along. Players choose between 1 and 10 numbers from the range 1 to \(80 ; 20\) winning numbers are displayed on a screen every four minutes. If you choose just one number, your probability of winning is \(20 / 80\), or \(0.25\). Lester plays one number eight times as he sits in a bar. What is the probability that all eight bets lose?

Peanut allergies among children. About \(2 \%\) of children in the United States are allergic to peanuts. \({ }^{21}\) Choose three children at random, and let the random variable \(X\) be the number in this sample who are allergic to peanuts. The possible values \(X\) can take are \(0,1,2\), and 3 . Make a three- stage tree diagram of the outcomes (allergic or not allergic) for the three individuals, and use it to find the probability distribution of \(X\).

Common Names. The U.S. Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez, and Wilson. These names account for \(9.6 \%\) of all U.S. residents. Out of curiosity, you look at the authors of the texthooks for your current courses. There are nine authors in all. Would you be surprised if none of the names of these authors were among the 10 most common? (Assume that authors' names are independent and follow the same probability distribution as the names of all residents.)

Lactose intolerance. Lactose intolerance causes difficulty digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States (ignoring other groups and people who consider themselves to belong to more than one race), \(82 \%\) of the population is white, \(14 \%\) is black, and \(4 \%\) is Asian. Moreover, \(15 \%\) of whites, \(70 \%\) of blacks, and \(90 \%\) of Asians are lactose intolerant. 22 (a) What percent of the entire population is lactose intolerant? (b) What percent of people who are lactose intolerant are Asian?
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