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Finding the probability of complementary events is the subject of this lesson

Subject:

Math  

Grade:

8  

 

Title – Probability
By – Ayman
Primary Subject – Math
Grade Level – Year 8 Stage 4

TOPIC/Outcome: Probability NS 4.4

YEAR/STAGE: Year 8, Stage 4

Outcomes:

      At the end of this lesson, students will be able to:

 

  • Identifying the complement of an event, e.g.’The complement of drawing a red card from a deck of cards is drawing a black card’.
  • Finding the probability of a complementary event

Pre-requisites:

  • Students must know that the sum of the probabilities in any situations must always add up to 1.
  • Express the probability of a particular outcome as a fraction between 0 and 1.
  • Perform basic numerical operations using fractions with different or equal numerators, decimals and percentages.
  • Convert between fractions, decimals and percentages.

Aids: Chalk, board, jars, and colored marbles

Teaching Strategy:

      Short Questions:

 

        Question 1:

          Calculate:

 

          1.   1 – 1/6

 

          2.   1/5 + 4/5

 

          3.   2/6 + 2/3

 

          4.   1 – 0.4

 

          5.   0.36 + 0.64

 

          6.   100% – 80%

 

        Question 2:

          Change the following:

 

          1.   0.56 to a fraction

 

          2.   1/6 to a decimal

 

          3.   1/4 to a percentage

 

          4.   0.78 to a percentage

 

          5.   75% to a fraction

 

          6.   65.6% to a decimal

 

      Comment- Advise students to check the work when they finish, and to attempt all questions. The first question aims to review performing operations with fractions, decimals and percentages, which the students will need when it comes to using the formula P(E) + P(Ē) = 1 or P(Ē) = 1 – P(E).
      (

Note from LessonPlansPage.com: If you see a box in the preceding formula, your browser does not accept that character. The character you should see can be described as E complement, or E with a macron, or E with a long vowel symbol above it.

      )
    Question 2 requires the students to perform conversion between fractions, decimals and percentages. If the students found any trouble with any of the above questions, I will give them another similar set after marking the questions.

Introduction:

      Today’s lesson is about complementary events, after we have discussed the difference between theoretical and experimental probabilities and learned how to change the probability to a percentage and decimal, we will discuss and find the probability of the complementary events.
    As mentioned in previous lessons the sum of probabilities in any situation must always add up to 1. For example, if the probability of winning a particular game is 1/x, then the probability of losing or not winning the game is 1 – 1/x. So the probability that the event does not occur is called the complement of that event.

Explication 1

      In this explication, I will make my students use general terms and not numerical values.
      State the complement of each event.

        a.   Tossing a head with a coin

 

        b.   Spinning a number that is greater than 10 on a wheel

 

        c.   Choosing a heart or a club from a pack of cards

 

        d.   Not winning a game of hockey

 

      Answers:

        e.   Tossing a tail with a coin

 

        f.   Spinning a number that is less than or equal to 10 on a wheel

 

        g.   Choosing a diamond or a spade from a pack of cards

 

        h.   Winning a game of hockey

 

    Comment – the purpose of this explication is to enable the students to understand the concept of complementary events.

Explication 2

      Divide the students into groups; provide each group with a jar and a set of colored marbles (6 blue, 7 black and 5 yellow marbles). In order to compare the answers we should have same number of marbles from each color with each group.
      All groups are to conduct this experiment; and are required to answer the following questions.
      Find the probability of drawing at random:

This can be calculated using the formula: number of favorable marbles / total number of marbles

        1.   a blue marble

 

        2.   a black marble

 

        3.   a yellow marble

 

        4.   not blue marble

 

        5.   not black marble

 

        6.   not yellow marble

 

      Ask the students to add the answers of 1 and 4, 2 and 5, 3 and 6. Hopefully, all the groups obtained 1.
      Answers:

        1.   6/18

 

        2.   7/18

 

        3.   5/18

 

        4.   12/18

 

        5.   11/18

 

        6.   13/18

 

      I shall explain to the students that the probability of an event E that occurs is P(E) and the probability that the event does not occur is P(Ē) .
      Ask the students to use for the probability of drawing a blue marble P( blue marble) and for the probability of drawing not a blue marble is P(not blue marble).
      Ask the students to write a relationship between P( blue marble) and P(not blue marble) based on the questions above.
    Comment- The students should have a common consensus that the probability of an event and its complement is always equal to 1.

Written Record:

      In your own words, explain what is meant by complementary events and how it could be calculated.
      Derive a general formula for the probability of complementary events using the general terms P(E) and P(Ē).
      Comment – I would expect them to explain its meaning and to derive the formulas that would be used to calculate the complementary events. All students are expected to produce this formula: P(E) + P(Ē) = 1 or P(Ē) = 1 – P(E).
    Now put this formula in a box so you can always remember it.

Worked Example:

      1.   A bag contains 9 marbles, 5 of which are red. Let A represent the event ‘drawing a red marble’. If one marble is drawn at random from the bag, find:

        a.   P(A)

 

        b.   P(Ā)

 

        c.   P(A) + P(Ā)

 

        as a fraction, a decimal [2 d.p.] and as a %.

      Solutions:

        a.   P(A) = P(red marble) = 5/9 = 0.56 = 56%

 

        b.   P(Ā) = P(not a red marble) = 1 – P(A) = 1 – 5/9 = 1/9 = 0.45 = 45%

 

        c.   P(A) + P(Ā) = 5/9 + 1/9 = 1

 

      Comment – This example should be worked out on the board with the help of the students; the next question will get all the students of the class involved.

Note from LessonPlansPage.com: If your browser displays a symbol box in the above equations, substitute the letter A complement, or A with a micron or A with a long vowel symbol above it.

      2.   The probability that a certain couple will have a child with green eyes is 1/4. What is the probability that the child will not have green eyes?
      Solution:

 

      P(not with green eyes) = 1 – P(with green eyes) =

 

      1 – 1/4 = 3/4.
    Comment – Ask the students to work out this question on their exercise book, then select one student to work it out on the board. Students are expected to start on the following activities by themselves.

Activity:

      Attempt the following questions:

 

      1.    The events E and F are complementary events. What is the value of P(E) + P(F)?

 

      2.    What can you say about the events A and B if

 

      P(A) + P(B) = 1?

 

      3.    The probability of winning a prize is 1/100. What is the probability of not winning a prize?
    Comment – The students will be given 3 or 4 minutes to attempt the above questions. I will ask three students to give me answers.

Homework:

      I will set questions similar to the questions discussed in the class.
      1.   A barrel contains 7 blue discs, 4 orange discs and 9 purple discs. Find, as a decimal, the probability that a disc drawn at random from the barrel is:

        a.    Purple

 

        b.    Blue

 

        c.    Not purple

 

        d.    Orange

 

        e.    Not blue

 

        f.    Not orange

 

      2.    The traffic lights at a certain intersection show red 45% of the time, amber 15% of the time and green the rest of the time. If I drive through the intersection, what is the probability that the lights will be:

        a.    Red?

 

        b.    Not green?

 

        c.    Green or amber?

 

      d.    Neither red nor green?

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