Topic: Counting and Probability
Class: Grade 12 (MDM4U)
Subject: Mathematics
Authors: Jessica Stone, Michelle King, Nakesha McKenzie
Date: November 17, 2010 / Length of Time
Important Notes
· Teachers’ dialogue with students during the lesson is written in italics
· Timed activities will be monitored using a timer / ------
Curriculum Expectations (taken directly from the Ontario Mathematics Curriculum, Grades 11 & 12)
“1.1: Recognize and describe how probabilities are used to represent the likelihood of a result of an experiment (e.g., spinning spinners; drawing blocks from a bag that contains different coloured blocks; playing a game with number cubes; playing Aboriginal stick-and-stone games) and the likelihood of a real-world event (e.g., that it will rain tomorrow, that an accident will occur, that a product will be defective)”
“1.6: Determine whether two events are independent or dependent and whether one event is conditional on another event, and solve related probability problems [e.g., calculate P(A and B), P(A or B), P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)” / ------
Related Learning Opportunities
Students will
· Calculate the probability of two or more independent events
· Calculate the probability of two or more dependent events
· Demonstrate the difference between an independent event and a dependent event / ------
Prior Knowledge
· Students may have limited knowledge of probability and how it relates to everyday life, except for when it comes to “common” examples (e.g. gambling)
· According to the grade 11 university Math curriculum, teaching probability is not an expectation, so the level of understanding and calculating probabilities will vary from student to student / ------
Prior Assumptions
· Students know that, in the case of two independent events, the probability of one event occurring is 1 out of 2 (i.e. “50-50”)
· Students are familiar with independent events, but may not be (as) familiar with dependent events and with finding the probability of dependent events
· Students might not realize that the chances of an event occurring is synonymous with the probability of the event occurring (i.e. unfamiliarity with terminology and concept) / ------
Resources
· Socks (blue, black, brown, white, and red)
· Bag (to put the socks in)
· Coins (enough for each group)
· Dice (one per group)
· Deck of cards
· Handout
· Chart paper
· Markers / ------
Agenda
1. Deal or No Deal?
2. Independent Events
3. Dependent Events
4. What’s the Deal?
5. Ticket Out the Door / ------
Planned Activities
Minds On Activity: Deal or No Deal?
Read the following challenge to students:
A magician produces a deck of four cards, 2 red and 2 black, and deals them face down [the backs of the cards are indistinguishable.]
He bets that you cannot point to two cards of the same color, and offers even odds.
· A friend urges you to take the bet:
"There are only three cases: 2 red, 2 black and one each.
You win 2 out of 3 times."
· Another friend says you will neither win nor lose:
"There are four cases: red-red, red-black, black-red, and black-black.
You win 2 out of 4 times, and that just matches the odds."
Make a choice (which of your friends is correct? How do you know?).
After students have had a chance to read the challenge, ask them to take a few minutes to choose the answer they believe is correct and go to the side of the classroom that corresponds to their choice (before class, three signs will be posted on designated walls of the classroom; one sign will say “Choice A – win 2 out of 3 times”, another will say “Choice B – win 2 out of 4 times” and the last will say “Choice C – Neither A or B”).
Students will be asked to discuss explanations for their choice with their group. After a few minutes, ask one or two people from each group to summarize their team’s explanation.
In the event that at least one of the areas remains empty, develop some potential reasons in support of the corresponding choices.
Afterward, ask students to return to their desk to begin the next part of the lesson. Take up the answer to this challenge question later on in the lesson (solution is written below in the next section).
This problem deals with probability, or the chance of, an event occurring. Today we will discuss two types of probabilities, beginning with the probability of independent events.
Lesson Part I – Probability Theory: Independent Events
Definition: / Two events, A and B, are independent if the fact that A occurring does not affect the probability of B occurring.
Some other examples of independent events are:
· Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
· Choosing a marble from a jar AND landing on heads after tossing a coin.
· Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.
· Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.
To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below. Note that multiplication is represented by the word “AND” (Note: this is not to be confused with the term “of”, which is used when dealing with fractions/percents; in that context, “of” implies multiplication as well).
Multiplication Rule 1: / When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)
Example (demonstrate in class with different coloured socks):
A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. The first pair you pull out is red --the wrong color. You replace this pair and choose another pair of socks. What is the probability that you will choose the red pair of socks twice?
There are a couple of things to note about this experiment. Since the first pair was replaced, choosing a red pair on the first try has no effect on the probability of choosing a red pair on the second try. Therefore, these events are independent.
Solution:
Probabilities:
P(red) / = / 1
5
P(red and red) / = / P(red) / · / P(red)
= / 1 / · / 1
5 / 5
= / 1
25
= 0.04 or 4%
Therefore, there is a 4% chance that I will choose the red socks twice.
Now try this example (students work in small groups and share the supplies):
Experiment 2: (Give each pair/small group a die and a coin) A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.
Solution:
Probabilities:
P(head) / = / 1
2
P(3) / = / 1
6
P(head and 3) / = / P(head) / · / P(3)
= / 1 / · / 1
2 / 6
= / 1
12
= 0.0833 or 8.33%
Therefore, the probability of landing on heads and rolling a 3 is 1/12, or 8.33%.
Now let’s take a look at the probability of dependent events.
Lesson Part II – Probability Theory: Dependent Events
Definition: / Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed.
To find the probability of the two dependent events, we use a modified version of Multiplication Rule 1, which was presented in the last lesson.
Multiplication Rule 2: / When two events, A and B, are dependent, the probability of both occurring is: P(A and B) = P(A) · P(B, given that A was removed)
Example (demonstration in class with cards):
A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? /
Analysis: The probability that the first card is a queen is 4 out of 52. However, if the first card is not replaced, then the second card is chosen from only 51 cards. Accordingly, the probability that the second card is a jack given that the first card is a queen is 4 out of 51. Conclusion: The outcome of choosing the first card has affected the outcome of choosing the second card, making the second event dependent on the first.
Solution:
Probabilities: / P(queen on first pick) / = / 4
52
P(jack on 2nd pick given queen on 1st pick) / = / 4
51
P(queen and jack) / = / 4 / · / 4 / = / 16 / = / 4
52 / 51 / 2652 / 663
= 0.006 or 0.6%
Therefore, the probability that the first card chosen is a queen and the second card is a jack, is 0.6%.
Now try this example (students may discuss with peers):
Experiment 2: Mr. Parietti needs two students to help him with a science demonstration for his class of 18 girls and 12 boys. He randomly chooses one student who comes to the front of the room. He then chooses a second student from those still seated. What is the probability that both students chosen are girls?
Solution:
Probabilities:
P(Girl 1 and Girl 2) / = / P(Girl 1) / and / P(Girl 2|Girl 1)
= / 18 / · / 17
30 / 29
= / 306
870
= / 51
145
= 0.35 or 35%
Therefore, there is a 35% chance that both students chosen are girls.
What’s the Deal?
· After each example, take up the answers as a class, provide clarification and answer questions from students.
· Then, ask students to return to their original side of the classroom they chose for during the Minds On activity.
· Give students some time to review the question and their answer(s) once more as a group. Invite students to use what they have learned in today’s lesson to solve the problem. Then, allow students to switch sides if they choose (last chance).
· Ask each group to explain/justify their answer:
Did anyone in your group change sides? Why or why not?
Reveal the answer (below) and go through the solution as a class.
Solution (Minds On):
Given: Four cards; 2 red, 2 black.
There are 3 possible outcomes: red & red, black & black, and red & black (Note: since order is not considered in this example, red & black is the same as black & red). Two of the three outcomes are “winning outcomes”, so the probability of winning is 2 out of 3, or 2/3. / 10 min
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Consolidation
Ticket Out the Door
Ask students the following question: If we changed the way the question from the beginning of the class was worded, does this change the outcome? For example:
A magician produces a deck of four cards, 2 red and 2 black, and deals them face down [the backs of the cards are indistinguishable.]
He bets that you cannot pick two cards of the same color. You pick the first card and it is red. What are your chances of picking another red card on your second pick (given you do not replace the red card you initially picked)?
Students will answer the question (i.e. explain how this changes the scenario and justify reasoning with calculations) on a blank sheet of paper. Collect the papers at the end of class (you may choose to have students write their names on their submission).
Solution:
Given: Four cards; 2 red, 2 black
The events are dependent. We know that the first card chosen is red and it is not replaced. So, you are left with 3 cards, two of which are black and one red. So, the probability of picking the second card to be red is 1 out of 3, or 1/3.
For students who finish early...
· Give them a challenge problem (e.g. could be a Fermi problem, or a problem related to next day’s lesson)
· Ask students with a strong grasp of the material to help students who might be struggling with the Ticket Out the Door question / 10 min
References
OAME. (2007). OAME Ontario Association For Mathematics Education. Retrieved November 10, 2010, from MDM4U Resources: http://www.oame.on.ca/main/files/Gr12-2007/MDM4U/MDM4U-unit1.pdf
Unknown. (2009, August). BrainDen.com. Retrieved November 10, 2010, from http://brainden.com/forum/index.php?/topic/9935-probability-challenge/
NOTE: Add a section for Accommodations/Modifications for lesson implementation