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Chapter 10: Problem 5

Draw and label a segment like this one. Plot and label points on your segment to represent the probability for each situation. a. You will eat breakfast tomorrow morning. b. It will rain or snow sometime during the next month in your hometown. c. You will be absent from school fewer than five days this school year. d. You will get an A on your next mathematics test. e. The next person to walk in the door will be under 30 years old. f. Next Monday every teacher at your school will give 100 free points to each student. g. Earth will rotate once on its axis in the next 24 hours.

Short Answer

Expert verified
Plot points for each event on a line from 0 (impossible) to 1 (certain): a ~0.9, b ~0.75, c ~0.85, d ~0.7-0.5, e ~0.8, f ~0.01, g ~0.99.

Step by step solution

01

Set Up the Probability Line Segment

Draw a horizontal line and label it from 0 to 1. This represents the probability scale where 0 is an impossible event, and 1 is a certain event.
02

Evaluate Probability for 'Eating Breakfast Tomorrow'

Decide the likelihood of eating breakfast tomorrow morning. Since eating breakfast is a common activity, you can label a point near 0.9 on the line, indicating high probability.
03

Determine Probability for 'Rain or Snow Next Month'

Consider local weather patterns for your hometown. If rain or snow is common, like in a temperate climate, mark your point around 0.75; if it's rare, choose a lower probability.
04

Assess Probability for 'Being Absent Fewer Than Five Days'

Most students attend school nearly every day, placing this scenario at a high probability, say around 0.85, assuming no major illnesses or events occur.
05

Calculate Probability for 'Getting an A on Next Math Test'

This depends on your prior math performance. If consistently excellent, place around 0.7; if unsure, choose around 0.5, denoting a moderate chance.
06

Evaluate Probability for 'Next Person Under 30'

Consider the typical age range of individuals around you. If you are in a high school setting, this probability could be high, around 0.8, given most teachers and students are under 30.
07

Assess Probability for 'Teachers Give Free Points'

This is an unlikely scenario and could be marked around 0.01, indicating a near impossibility.
08

Determine Probability for 'Earth Rotating in 24 Hours'

The Earth's rotation is a natural, inevitable event, so this probability should be very close to 1, around 0.99.

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Key Concepts

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

Probability Line Segment
To understand probability, we often visualize it on a line segment. This line runs horizontally and is divided into increments between 0 and 1. Each number on this segment represents a possible probability of an event occurring. A probability of 0 means the event is impossible, while a 1 means it is certain to happen.
Points in between these two extremes show varying likelihoods. For example, drawing a point at 0.5 reflects an equal chance of the event happening or not happening. By mapping probabilities onto this line segment, we gain a clearer visual representation of how likely different events are.
Probability Scale
The probability scale is a crucial element when discussing various scenarios. This scale runs from 0 to 1, just like the probability line segment. However, it serves as a tool to measure and compare the likelihood of events more fluidly.
  • A score of 0 means the event won't occur under any circumstances.
  • A middle value, like 0.5, indicates even odds.
  • A value near 1 means an event is almost certain.
Understanding this scale helps us conceptualize everyday situations. For example, if we say the probability of rain tomorrow is 0.3, this shows rain is not expected, but conditions might still allow for it.
Probability Scenarios
When handling probability scenarios, we often evaluate everyday events and assign probabilities to them by considering frequency and conditions. For instance, when considering if you will eat breakfast tomorrow, you might rate this as highly probable if it's part of your regular routine.
  • Regular events, like eating breakfast, often sit at higher probabilities like 0.9.
  • Unlikely events, such as all teachers giving out 100 free points, depict near-zero probabilities around 0.01.
  • Certainties, like Earth's rotation, are firmly at 0.99 or 1.
By assigning these probabilities, you understand how probable or improbable certain events are based on real-life contexts.
Educational Probability Exercises
Educational probability exercises are instrumental in building a robust understanding of probabilities. By plotting situations on a probability line segment, students can see how often events might occur.
Such exercises prompt students to investigate the likelihood of various scenarios and make educated guesses grounded in logic and past experiences.
  • These activities boost critical thinking by correlating agreements with numerical probability.
  • They help clarify misconceptions surrounding chance and inevitability.
  • They incorporate everyday scenarios, making lessons more relatable and practical.
Practicing with these types of exercises prepares students to analyze future events confidently and apply probability concepts in real-world situations.

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Most popular questions from this chapter

Identify each situation as a permutation, a combination, or neither. If neither, explain why. a. The number of different committees of 10 students that can be chosen from the 50 members of the freshman class. (a) b. The number of different ice-cream cones if all three scoops are different flavors and a cone with vanilla, strawberry, then chocolate is different from a cone with vanilla, chocolate, then strawberry. c. The number of different ice-cream cones if all three scoops are different flavors and a cone with vanilla, chocolate, then strawberry is considered the same as a cone with vanilla, strawberry, then chocolate. d. The number of different three-scoop ice-cream cones if you can choose multiple scoops of the same flavor.

Create a tree diagram with probabilities showing outcomes when drawing two marbles without replacement from a bag containing one blue and two red marbles. (You do not replace the first marble drawn from the bag before drawing the second.) (a)

Astrid works as an intern in a windmill park in Holland. She has learned that the anemometer, which measures wind speed, gives off electrical pulses and that the pulses are counted each second. The ratio of pulses per second to wind speed in meters per second is always \(4.5\) to 1 . a. If the wind speed is 40 meters per second, how many pulses per second should the anemometer be giving off? b. If the anemometer is giving off 84 pulses per second, what is the wind speed?

Suppose there are 180 twelfth graders in your school, and the school records show that 74 of them will be attending college outside their home state. You conduct a survey of 50 twelfth graders, and 15 tell you that they will be leaving the state to attend college. What is the theoretical probability that a random twelfth grader will be leaving the state to attend college? Based on your survey results, what is the experimental probability? What could explain the difference?

In April 2004 , the faculty at Princeton University voted that each department could give A grades to no more than \(35 \%\) of their students. Japanese teacher Kyoko Loetscher felt that 11 of her 20 students deserved A's, as they had earned better than \(90 \%\) in the course. However, she could give A's to only \(35 \%\) of her students. How many students is this? Draw two relative frequency circle graphs: one that shows the grades (A's versus non-A's) that Loetscher would like to give and one that shows the grades she is allowed to give. (Newsweek, February 14, 2005, p. 8) (a)
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