probability

Introduction

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PROBABILITY

Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between 1 and 0. An event with a probability of 1 can be considered a certainty: for example, the probability of a coin toss resulting in either "heads" or "tails" is 1, because there are no other options, assuming the coin lands flat. An event with a probability of .5 can be considered to have equal odds of occurring or not occurring: for example, the probability of a coin toss resulting in "heads" is .5, because the toss is equally as likely to result in "tails." An event with a probability of 0 can be considered an impossibility: for example, the probability that the coin will land (flat) without either side facing up is 0, because either "heads" or "tails" must be facing up. A little paradoxical, probability theory applies precise calculations to quantify uncertain measures of random events.

To introduce probability theory through simple experiments.To use the formula for finding the probability of an event.To find the probabilities of events with equally likely and non-equally likely outcomes.

Image result for importance of probability

 

 

Task

Objectives:

in the end of the lesson, the learners should able to:

1. learn what probability is,
2. learn different ways to express probability numerically: as a ratio, a decimal, and a percentage, and
3. learn how to solve problems based on probability.

 

 

 

Process

1. Begin the lesson by asking students to define probability (the likelihood or chance that a given event will occur). Probability is usually expressed as a ratio of the number of likely outcomes compared with the total number of outcomes possible. Ask students if they can give an example of probability.
2. To help students understand probability, work on the following problem as a class: Imagine that you have boarded an airplane. The rows are numbered from 1 to 30, and there are six seats per row, three on each side of the isle. Seats in each row are labeled A through F. Using that information, work together as a class to solve the problems listed below.

  1. How many seats are in the airplane? 180 seats
  2. What are your chances of sitting in row number 7? 6/180, or 1/30
  3. What are your chances of sitting in a window seat? There are two window seats per aisle, for a total of 60 window seats. Your chances of seating at a window would be 60/180, or 1/3.
  4. What are your chances of sitting in an "A" seat? There are 30 A seats, so your chances are 30/180, or 1/6.
  5. What are your chances of sitting in an even-numbered row? Of the 30 rows, 15 are even-numbered, so your chances are 15/30, or 1/2.
3.

To figure out each problem, students must set up a ratio between the total number of outcomes—in these problems either the total number of seats or rows—and the specific question asked. Tell students that they will write their answer as a fraction, decimal, and percentage. Example: The chance of sitting in seat 7A is 1/180, .00555, or .555 percent. The ratio presented as a percentage helps make it clear if the probability of an event is great or small.

ACTIVITY 1

  1. what is probability?
  2. give an example that we can use probability?
  3. What is the probability of getting a sum 9 from two throws of a dice?                                                       A. 1/6  B. 1/8  C. 1/9  D. 1/12
  4. Three unbiased coins are tossed. What is the probability of getting at most two head                             A. 3/4  B.1/4   C. 3/8  D. 7/8

  5. A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?                                                                                                                         A. 3/4  B. 4/7  C. 1/8  D. 3/7

  • ACTIVITY 2

1. Imagine you are on the school debate team and the subject at hand is whether companies should drill for oil in Antarctica. What statistics would you look for if you're arguing in favor of oil exploration there? What statistics would you look for to support your argument against drilling there? What are some ways that numbers could support arguments on both sides?
2. Think about numbers you may have seen in advertisements, such as "X Juice is 90 percent real juice," or "Y cereal has 35 percent of the total vitamins needed in one day." How would you write each percentage as a ratio?
3. What does it mean when you hear the weather reporter predict a 10 percent chance of rain? Is that a high or low probability?
4. Express the probability that your mother will let you have a sleepover next weekend as a probability, assuming that the total number of outcomes is 100. What factors would increase the probability that she would say yes? (If you finish all your homework and chores, go to bed on time.) What factors would decrease the probability that she would say yes? (If you misbehave, do not finish your homework or chores, or go to bed on time.)
5.

How do you think authors of The Farmer's Almanac make their predictions about weather for a year? How do you think they use probability?

Evaluation

Use percentage point to evaluate students' work during this lesson:

  • 90%-100%: demonstrates a strong understanding of probability based on their participation in class, their ability to complete the Classroom Activity Sheet, and their ability to complete the Take-Home Activity Sheet
  • 80%-90%: demonstrates a moderate understanding of probability based on their participation in class, their ability to complete the Classroom Activity Sheet, and their ability to complete the Take-Home Activity Sheet
  • 75%-80%: demonstrates a weak understanding of probability based on their participation in class, their ability to complete the Classroom Activity Sheet, and their ability to complete the Take-Home Activity Sheet.

Conclusion

 

  • Probability 

    In general:

    Probability of an event happening = Number of ways it can happen Total number of outcomes

     

    Example: the chances of rolling a "4" with a die

    Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

    Total number of outcomes: 6 (there are 6 faces altogether)

    So the probability = 1 6

    Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

    Number of ways it can happen: 4 (there are 4 blues)

    Total number of outcomes: 5 (there are 5 marbles in total)

    So the probability = 4 5 = 0.8

Probability in Advertising

Ask students to look at newspapers and magazines for examples of how numbers are used in advertisements. For example, it is not unusual to see something like "two-thirds less fat than the other leading brand" or "four out of five dentists recommend Brand T gum for their patients who chew gum." Why do advertisers use numbers like these? What information are they trying to convey? Do students think that the numbers give accurate information about a product? Why or why not?



They Said What?

Ask students to look at newspapers or magazines for examples of how politicians, educators, environmentalists, or others use data such as statistics and probability. Then have them analyze the use of the information. Why did the person use data? What points were effectively made? Were the data useful? Did the data strengthen the argument? Have students provide evidence to support their ideas.

Credits