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Chapter 9: Problem 33

The probability of winning the grand prize at a particular carnival game is 0.005. Is the outcome of winning very likely or very unlikely?

Short Answer

Expert verified
Winning the grand prize is very unlikely, with a probability of 0.005.

Step by step solution

01

Understanding the Probability Scale

Probabilities range from 0 to 1, where 0 indicates an impossible event and 1 indicates a certain event. A probability closer to 0 suggests that an event is very unlikely, whereas a probability closer to 1 suggests that an event is very likely.
02

Interpreting Small Probabilities

A probability of 0.005 is quite small, meaning it is very close to 0 on the probability scale. This indicates that the occurrence of this event (winning the grand prize) is not very likely and is, in fact, quite rare.
03

Comparing to Thresholds

Typically, events with probabilities less than 0.05 (or 5%) are considered statistically rare or unlikely. Since 0.005 is much less than 0.05, it reinforces the conclusion that winning is very unlikely.

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Key Concepts

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

Probability Scale
Understanding the probability scale is essential when working with probabilities, as it helps you to determine how likely an event is to occur. The scale ranges from 0 to 1. A probability of 0 means that an event is impossible, like trying to roll a 7 on a standard six-sided die.
On the other hand, a probability of 1 means an event is certain, like the sun rising in the east.
Most probabilities fall somewhere between these two extremes.
  • A probability closer to 0 indicates that an event is less likely to occur, or unlikely.
  • A probability closer to 1 indicates that an event is more likely to happen, or likely.
  • A probability of 0.5 implies an event is equally likely to happen as not happen, like flipping a fair coin and getting heads.
The scale is useful for visualizing and comparing the likelihood of different events. This helps you make informed decisions based on how probable an event is estimated to be.
Event Likelihood
The likelihood of an event occurring can be thought of as the chance or probability of its occurrence. To determine if an event is likely or unlikely, compare its probability to established thresholds.

For example:
  • A probability less than 0.05 is generally considered rare or unlikely.
  • A probability between 0.05 and 0.5 is considered possible but not guaranteed.
  • A probability greater than 0.5 and less than 1 suggests an event is likely but not certain.
In the context of the given probability of 0.005, it is evident this is a rare occurrence since it is far less than 0.05. Therefore, it implies that winning the grand prize in the carnival game is very unlikely.
Interpreting Probabilities
Interpreting probabilities involves understanding what a given probability means in terms of real-world expected outcomes. It is crucial for making predictions and informed decisions.

A few tips for successfully interpreting probabilities are:
  • Recognize that probabilities provide a number of how often an event is expected to occur in a large number of trials.
  • Remember that a small probability indicates the event might happen sometimes, but not often.
  • Understand that probabilities near 0.5 mean an event is as likely to happen as not.
  • Compare the probability to known thresholds to assess the rarity or likelihood of an event, such as using 0.05 or 5% when determining if something is rare.
By interpreting the probability of 0.005, one can conclude that an event, like winning a carnival game, is exceptionally rare and unlikely to occur often, if at all. Proper interpretation helps in setting realistic expectations about outcomes.

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Most popular questions from this chapter

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are: a. \(H_{0} : x=4.5, H_{a} : \quad x > 4.5\) b. \(H_{0} : \mu \geq 4.5, H_{a} : \mu < 4.5\) c. \(H_{0} : \mu=4.75, H_{a} : \mu > 4.75\) d. \(H_{0} : \mu=4.5, H a : \mu > 4.5\)

Assume \(H_{0} : p=0.25\) and \(H a : p \neq 0.25 .\) Is this a left-tailed, right-tailed, or two-tailed test?

"Macaroni and Cheese, please!!" by Nedda Misherghi and Rachelle Hall As a poor starving student I don't have much money to spend for even the bare necessities. So my favorite and main staple food is macaroni and cheese. It's high in taste and low in cost and nutritional value. One day, as I sat down to determine the meaning of life, I got a serious craving for this, oh, so important, food of my life. So I went down the street to Great way to get a box of macaroni and cheese, but it was SO expensive! 2.02 dollar !!! Can you believe it? It made me stop and think. The world is changing fast. I had thought that the mean cost of a box (the normal size, not some super-gigantic- family-value-pack) was at most 1 dollar, but now I wasn't so sure. However, I was determined to find out. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. Here are the data I wrote in my notebook: Price per box of Mac and Cheese: \- 5 stores @ 2.02 dollar \- 15 stores @ 0.25 dollar \- 3 stores @ 1.29 dollar \- 6 stores @ 0.35 dollar \- 4 stores @ 2.27 dollar \- 7 stores @ 1.50 \- 5 stores @ 1.89 dollar \- 8 stores @ 0.75 . I could see that the cost varied but I had to sit down to figure out whether or not I was right. If it does turn out that this mouth-watering dish is at most 1 dollar, then I'll throw a big cheesy party in our next statistics lab, with enough macaroni and cheese for just me. (After all, as a poor starving student I can't be expected to feed our class of animals!)

Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal. Is this a test of means or proportions?

Which distribution do you use when you are testing a population mean and the standard deviation is known? Assume sample size is large.
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