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Chapter 6: Problem 6

"N.Y. Lottery Numbers Come Up \(9-1-1\) on \(9 / 11^{*}\) was the headline of an article that appeared in the San Francisco Chronicle (September 13,2002\()\). More than 5600 people had selected the sequence \(9-1-1\) on that date, many more than is typical for that sequence. A professor at the University of Buffalo is quoted as saying, "Tm a bit surprised, but I wouldn't characterize it as bizarre. It's randomness. Every number has the same chance of coming up. People tend to read into these things. I'm sure that whatever numbers come up tonight, they will have some special meaning to someone, somewhere." The New York state lottery uses balls numbered \(0-9\) circulating in three separate bins. To select the winning sequence, one ball is chosen at random from each bin. What is the probability that the sequence \(9-1-1\) would be the one selected on any particular day?

Short Answer

Expert verified
The probability that the sequence 9-1-1 would be the one selected on any particular day is \(\frac{1}{1000}\).

Step by step solution

01

Identifying the Total Number of Outcomes

The selection process involved 3 bins with 10 balls numbered from 0 to 9 in each bin. Therefore, the total number of outcomes when picking a ball from each bin is given by \(10 \times 10 \times 10\), which is 1000. This is because for every single ball in the first bin, there are 10 possible counterparts in the second and 10 more in the third bin.
02

Identifying the Number of Favorable Outcomes

We're asked to calculate the probability of choosing the sequence 9-1-1. There is only one occurrence of this exact sequence. So, the number of favorable outcomes is 1.
03

Calculating the Probability

Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. So, the probability of the sequence 9-1-1 is given by \(\frac{1}{1000}\).

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Key Concepts

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

Randomness
Randomness refers to the lack of any predictable order or plan in events. In probability, this is essential because it means every event has an equal chance of occurring, independent of past outcomes. For example, in a lottery system, randomness ensures that no particular sequence is more likely than another. This unpredictability is key to fairness and forms the foundation for calculating probabilities. Without randomness, outcomes could be manipulated or predicted, which would undermine the entire system of probability used in making statistical inferences.
Lottery probability
In the context of a lottery, probability is a way to measure how likely it is for a specific set of numbers to be drawn. Imagine the process of selecting winning numbers as a game of chance, where each number combination from 000 to 999 has the same likelihood of being selected. Every sequence, whether it鈥檚 123, 456, or 911, has an equal chance, specifically, a probability of \(\frac{1}{1000}\) in a three-number lottery system. This occurs because each of the 10 digits (0-9) in each bin is equally likely and chosen independently.

  • In lotteries, each unique combination is a distinct event.
  • Probabilities help manage expectations and understand odds.
  • The goal is to ensure a fair and unbiased method for everyone participating.
Favorable outcomes
Favorable outcomes refer to the specific event or outcome that aligns with what you're hoping for in a probability scenario. In the provided example, the sequence 9-1-1 is the favorable outcome. This is the unique combination of numbers that, if chosen, corresponds to winning or achieving a desired result. The probability of a favorable outcome depends on how many such successful outcomes are possible out of all possible outcomes.

In a lottery draw:
  • Each sequence, like 9-1-1, is considered a favorable outcome if it's your chosen number.
  • Even with the massive total combinations, each single chosen combination stands alone as favorable.
  • Achieving a favorable outcome is purely due to chance in a random lottery draw.
Total number of outcomes
The total number of outcomes in any probability experiment is the full set of possible results that could occur. For a lottery where each number is picked from a set of ten (0 through 9) and there are three positions to fill, the total number of outcomes is calculated as \(10 \times 10 \times 10 = 1000\). This formula represents all the possible combinations that can be drawn from the lottery.
  • Total outcomes are crucial for calculating probability; they form the denominator.
  • Understanding the full breadth of possibilities helps appreciate the odds of any single event.
  • It highlights why winning numbers are rare and valuable due to the enormous possible variations.
Each individual outcome among these 1000 possibilities is equally likely, reflecting the fairness and randomness in the lottery draw system.

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Most popular questions from this chapter

Is ultrasound a reliable method for determining the gender of an unborn baby? Consider the following data on 1000 births, which are consistent with summary values that appeared in the online journal of Statistics Education ("New Approaches to Learning Probability in the First Statistics Course" 120011 ): \begin{tabular}{l|cc} & Ultrasound Predicted Female & Utrasound Predicted Male \\ \hline Actual Gender Is Female & 432 & 48 \\ Actual Gender is Male & 130 & 390 \\ \hline \end{tabular} Do you think that a prediction that a baby is male and a prediction that a baby is female are equally reliable? Explain, using the information in the table to calculate estimates of any probabilities that are relevant to your conclusion.

The report "TV Drama/Comedy Viewers and Health Information" (www.cdc.gov/healthmarketing) describes the results of a large survey involving approximately 3500 people that was conducted for the Centers for Disease Control. The sample was selected in a way that the Centers for Disease Control believed would result in a sample that was representative of adult Americans, One question on the survey asked respondents if they had learned something new about a health issue or disease from a TV show in the previous 6 months. Consider the following outcomes: \(L=\) outcome that a randomly selected adult American reports learning something new about a health issue or disease from a TV show in the previous 6 months and \(F=\) outcome that a randomly selected adult American is female Data from the survey were used to estimate the following probabilities: $$ P(L)=.58 \quad P(F)=.50 \quad P(L \text { and } F)=.31 $$ Are the outcomes \(L\) and \(F\) independent? Use probabilities to justify your answer.

Many fire stations handle emergency calls for medical assistance as well as calls requesting firefighting equipment. A particular station says that the probability that an incoming call is for medical assistance is .85. This can be expressed as \(P(\) call is for medical assistance \()=.85 .\) a. Give a relative frequency interpretation of the given probability. b. What is the probability that a call is not for medical assistance? c. Assuming that successive calls are independent of one another (i.e., knowing that one call is for medical assistance doesn't influence our assessment of the probability that the next call will be for medical assistance), calculate the probability that both of two successive calls will be for medical assistance. d. Still assuming independence, calculate the probability that for two successive calls, the first is for medical assistance and the second is not for medical assistance. e. Still assuming independence, calculate the probability that exactly one of the next two calls will be for medical assistance. (Hint: There are two different possibilities that you should consider. The one call for medical assistance might be the first call, or it might be the second call.) f. Do you think it is reasonable to assume that the requests made in successive calls are independent? Explain.

The paper "Predictors of Complementary Therapy Use among Asthma Patients Results of a Primary Care Survey"" (Health and Social Care in the Community 12008 h \(155-164\) ) included the accompanying rable. The table summarizes the responses given by 1077 asthma patients to two questions: Question 1: Do conventional asthma medications usually help your asthma symptoms? Question 2: Do you use complementary therapies (such as herbs, acupuncture, aroma therapy) in the treatment of your asthma? \begin{tabular}{l|cc} & Doesn't Use Complementary Therapies & Does Use Complementary Theraples \\ \hline Comventional Medica- & 816 & 131 \\ \multicolumn{1}{c|} { tions Usually Help } \\ Conventional Medi- & & \\ cations Usually Do Not Help & 103 & 27 \\ \hline \end{tabular} From this information, we can estimate that the proportion who use complementary therapies is \(\frac{131+27}{1077}=\frac{158}{1077}=.147 .\) For those who report that conventional medications usually help, the proportion who use complementary therapies is \(\frac{131}{947}=.138\) and for those who report that conventional medications usually do not help, the proportion who use comple- mentary therapies is \(\frac{27}{130}=.208\) If one of these 1077 patients is selected at random, are the outcomes selected patient reports that conventional medications wually help and selected patient uses complementary therapies independent or dependent? Explain.

\quad\( + Approximately \)30 \%$ of the calls to an airline reservation phone line result in a reservation being made. a. Suppose that an operator handles 10 calls. What is the probability that none of the 10 calls results in a reservation? b. What assumption did you make in order to calculate the probability in Part (a)? c. What is the probability that at least one call results in a reservation being made?
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