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Chapter 13: Problem 28

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?

Short Answer

Expert verified
The probability that Lester loses all eight bets is approximately 0.1001.

Step by step solution

01

Understand the Problem

We need to find the probability that Lester loses all eight times he places a bet on a single number. Since the probability of winning with one number is 0.25, the probability of losing with one number is 1 - 0.25 = 0.75.
02

Calculate the Probability of Losing Once

The probability that Lester loses once is the complement of the probability that he wins. Therefore, the probability that he loses once is 0.75.
03

Calculate the Probability of Losing Eight Times

To find the probability of losing all eight times, we use the multiplication rule for independent events. The probability that he loses eight times in a row is calculated as \((0.75)^8\).
04

Compute the Result

Calculate \( (0.75)^8 \) to get the probability that Lester loses all eight bets.
05

Calculate Numerical Value

Computing \( (0.75)^8 \) gives approximately 0.1001.

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Key Concepts

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

Independent Events
In probability, independent events are acts or occurrences that have no effect on the likelihood of each other. This means that the outcome of one event does not alter the probability of another. Understanding independent events is vital because it allows us to calculate probabilities in scenarios where multiple events happen in sequence.
For example, in Lester's lottery play, each game of Quick Draw he participates in is independent of the others. His chance of winning or losing remains the same, regardless of the outcome of previous bets. This assumption of independence is key in statistical calculations when considering repeated trials of the same event.
  • Two events are independent if the occurrence of one does not change the probability of the other.
  • In repeated independent trials, like Lester's eight bets, past results do not influence future outcomes.
Such understanding allows us to apply certain probability rules to independently repeated events, which we'll discuss next.
Multiplication Rule
The multiplication rule of probability is used when we want to find the probability of two or more independent events occurring in sequence. According to this rule, the probability of all events occurring is the product of their individual probabilities.
For Lester's scenario, he places a bet eight times. To find the probability that he loses all eight times, we multiply the probability of losing each individual bet: \[(0.75) \times (0.75) \times (0.75) \times ... \text{(8 times)} = (0.75)^8\.\]
This calculation stems from the multiplication rule, which states:
  • If several events are independent, the probability of all occurring is the product of each probability.
  • This rule is practical for calculating the compound probability of sequential events.
The multiplication rule simplifies these calculations significantly and is foundational in solving problems involving series of independent trials.
Complement Rule
In probability, the complement rule is a simple yet powerful concept used to determine the probability of an event not happening. When you know the probability of an event, the complement rule allows you to find the probability of the event's non-occurrence by subtracting the known probability from 1.
In the exercise at hand, Lester's probability of winning one bet is 0.25. Therefore, by applying the complement rule:
  • The probability of losing is calculated as: \(1 - 0.25 = 0.75\).
  • Here, 0.75 represents the likelihood of one bet ending in a loss.
This rule is particularly useful when working with situations like Lester's. By understanding what it means not to achieve the desired outcome, we quickly solve for the probability of the opposite outcome. Here, knowing the probability of not winning allows us to move forward and use other concepts, like the multiplication rule, to tackle more complex probability questions.

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Most popular questions from this chapter

Universal Blood Donors. People with type O-negative blood are referred to as universal donors, although if you give type O-negative blood to any patient, you run the risk of a transfusion reaction due to certain antibodies present in the blood. However, any patient can receive a transfusion of O-negative red blood cells. Only 7.2\% of the American population have O-negative blood. If 10 people appear at random to give blood, what is the probability that at least one of them is a universal donor?

At the Gym. Suppose that \(8 \%\) of adults belong to health clubs, and \(45 \%\) of these health club members go to the club at least twice a week. What percentage 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.

Are They Independent? For each of the following pair of events \(A\) and \(B\), do you think the events are independent? Explain your reasoning. a. You select an adult U.S. citizen at random. Event \(A\) is "the person is a registered democrat," and event \(B\) is "the person is opposed to the death penalty." b. You select an adult U.S. citizen at random. Event \(A\) is "the person is a baby boomer" (baby boomers are people born 1946-1964), and event \(B\) is "the person favors legalization of marijuana." c. You draw a card at random from a deck of playing cards. Event \(A\) is "the card is the king of hearts," and event \(B\) is "the card is the queen of diamonds." d. You select a student at your college at random. Event \(A\) is "the student is taking a Spanish class," and event \(B\) is "the student has visited Mexico."

College Degrees. Of all postsecondary degrees awarded in the United States, including master's and doctorate degrees, \(21 \%\) are associate's degrees, \(58 \%\) are earned by people whose race is White, and \(12 \%\) are associate's degrees earned by Whites. \(-\) Make a Venn diagram and use it to answer these questions. a. What percentage of all degrees are associate's degrees earned by non- Whites? b. What percentage of non-Whites earn a college degree other than an associate's degree? c. What percentage of all degrees are earned by nonWhites?

.Student Debt. At the end of 2016 , the average outstanding student debt for bachelor's degree recipients was \(\$ 28,500\). Here is the distribution of outstanding education debt (in thousands of dollars): 17 \begin{tabular}{|l|c|c|c|c|c|c|} \hline Debt & \(<10\) & 10 to \(<20\) & 20 to \(<30\) & 30 to \(<40\) & 40 to \(<50\) & \(\geq 50\) \\ \hline Probability & \(0.40\) & \(0.13\) & \(0.17\) & \(0.12\) & \(0.08\) & \(0.10\) \\ \hline \end{tabular} a. What is the probability that a randomly chosen student has an outstanding debt of \(\$ 20,000\) or more? b. Given that a student has an outstanding debt of at least \(\$ 20,000\), what is the conditional probability that the debt is at least \(\$ 50,000 ?\)
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