CLIL - Lesson plan . Subject: Maths.
Title: Probability
Grade level: 2th Upper secondary school Author: Giulio Prevedello
Content : Introduction to the meaning of the probability.. Teaching aims: Probability scale. Probability that it will not happen happen (complementary probability) Experimental probability Theoretical probability Conditional probability. probability. Learning Outcomes Know: what is probability and objectives Be able to: calculate the probability of an event. Be aware: solving probabilistic problems They are writing on the board so that learners are clear about what they should achieve by the end of the lessons. Assessment
Content and language: Summative (standard test) and formative (informal) assessment.
Can the learners define probability Calculate experimental probability, probability, theoretical probability and conditional probability. probability. Solve problems with probability. probability. Communication
Examples of communication activities: Tell the learners they have 5 minutes to work with a partner to explain “What is probability?”
Solve with partner, simple problems proposed proposed as: for example:
“Each of the letters of the word MISSISSIPPI are written on separate pieces of paper that are then folded, put in a hat, and mixed thoroughly thoroughly.. One piece of paper is chosen (without looking) from the hat. What is the probability it is an I?”
. “A fair coin is tossed three times. What is the probability of obtaining one 1/16
Head and two Tails? (A fair coin is one that is not loaded, so there is an equal chance of it landing Heads up or Tails up.)” Cognition
Examples of cognition: meaning of complementary events; meaning of independent events in probability; meaning of conditional probability
Thinking Skills
use concepts of probability in real-life problems
Citizenship
Example of citizenship: Abrahm de Moivre, Karl Gauss and “Bell curve”
Procedure
Activate prior knowledge task
Language of learning
Event single observation in a probability experiment Likely probable Impossible an event is impossible if it can never happen Certain an event is certain if it will definitely happen Chance the likelihood or probability, of an event happening Trial a single observation in a probability experiment Sample Point just one of the possible outcomes Sample Space all the possible outcomes of an experiment Toss the coin Fair coin a ideal coin.(with probability 1/2 of success on each trial is metaphorically called a fair coin) Biased coin If a cheat has altered a coin to prefer one side over another Replace: Put it back in.. Outcome: Something that comes out of or follows from an activity or process; consequence Dependent Events: if the outcome of one event affects the outcome of another
Resources
Use an online dictionary with an audio function to hear maths vocabulary pronounced, e.g. http://www.amathsdictionaryforkids.com/dictionary.html http://www.languageguide.org/im/numbers/eng/extra4.jsp?lang=it Use a grammar reference book in order to practise producing complex sentences such as conditional forms, e.g. If you multiply x by 10 , you’ll see that __________.
Mathematics – C. Meyrick, J. Roberts –Oxford (Content and Language support) http://www.mathsisfun.com/data/index.html#stats http://www.mathsisfun.com/data/standard-normal-distribution.html
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Subject : Maths
Title – Probability
“This morning I’m going to talk about probability…. Main points -what is probability -calculate the probability of an event: theoretical probability, experimental probability. -probability of the complementary event; - conditional probability
Step-By-Step Procedures:
Lesson 1
Probability question How likely something is to happen. Many events can't be predicted with total certainty. The best we can do, is say how likely they are to happen, using the idea of probability.
Tossing a Coin When a coin is tossed, there are two possible outcomes:
heads (H) or
tails (T)
We say that the probability of the coin landing H is ½. (a half) And the probability of the coin landing T is ½.
Throwing Dice When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.
The probability of any one of them is 1/6.
Probability of an event happening = Number of ways it can happen
3/16
------------------------------------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)
1 So the probability = -6
Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble will be picked? Number of ways it can happen: 4 (there are 4 blues) Total number of outcomes: 5 (there are 5 marbles in total)
4 So the probability = -- = 0.8 5
4/16
Probability Line You can show probability on a Probability Line:
Probability is always between 0 and 1
Probability is Just a Guide Probability does not tell us exactly what will happen, it is just a guide
Probabilità defition: Activity (GCSE bitesize) Test More (uk)
Example: toss a coin 100 times, how many Heads will come up? Probability says that heads have a ½ chance, so we would expect 50 Heads. But when you actually try it out you might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.
Theoretical probability
For theoretical probability the calculations is: p ( A)
n
n= Number of ways it can happen
T
T= Total number of outcomes .
Example: “A die is thrown once. What is the probability that the score is a factor of 6?” 5/16
The factors of six are 1, 2, 3 and 6, so the Number of ways it can happen = 4 There are six possible scores when a die is thrown, so the Total number of outcomes = 6 So the probability that the score is a factor of six = 4/6 = 2/3
Video….. Experimental probability p ( A)
s
T
s=number of successful trials T=number of trials.
Example: Alex decide to see how many times a "double" would come up when throwing 2 dice. Each time Alex throws the 2 dice is an Experiment. It is an Experiment because the result is uncertain.
The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points: {1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}
The Sample Space is all possible outcomes (36 Sample Points): {1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}
These are Alex's Results: Experiment {3,4} {5,1} {2,2} {6,3} ...
Is it a Double? No No Yes No ...
After 100 Experiments, Alex had 19 "double" Events ... is that close to what you would expect?
6/16
An Experiment with a Die Throw a die 60 times, record the scores in a tally table.
You can record the results in this table using tally marks: Score
Tally
Frequency
1 2 3 4 5 6 Total Frequency = OK, Go! ... ...
Finished ...?
Now draw a bar graph to illustrate your results. You can fill in this one: Or you can use Data Graphs (Bar, Line and Pie) then print it out.
You may get something like this:
Are the bars all the same height?
If not ... why not?
7/16
60
60 Throws Why did I ask you to make 60 throws? Well, only 6 throws would not give you good results, 600 throws would have been too hard, so I chose 60, which is 10 lots of 6. So we should expect 10 of each number, like this:
Those are the theoretical values, as opposed to the experimental ones you got from your experiment! Compares the experimental results with theoretical results. This graph and your graph should be similar. If you did the experiment a very large number of times, you would get results much closer to the theoretical ones.
Questions
Which face came up most often? ____
Which face came up least often? ____
Do you think you would get the same results if you did this again? Yes / No
An experiment gives results. When done again it may give different results!
Probability: Complement Complement of an Event: All outcomes that are NOT the event.
When the event is Heads, the complement is Tails
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When the event is {Monday, Wednesday} the complement is {Tuesday, Thursday, Friday, Saturday, Sunday}
When the event is {Hearts} the complement is {Spades, Clubs, Diamonds, Jokers}
So the Complement of an event is all the other outcomes. And together the Event and its Complement make all possible outcomes.
The probability of an event is shown using "P": P(A) means "Probability of Event A"
The complement is shown by a little ' mark such as A' (or sometimes Ac or A): P(A') means "Probability of the complement of Event A"
The two probabilities always add to 1 P(A) + P(A') = 1
Example: Rolling a "5" or "6" Event A: {5, 6}
Number of ways it can happen: 2 Total number of outcomes: 6
2 P(A) =
1 =
6
3
4
2
The Complement of Event A is {1, 2, 3, 4} Number of ways it can happen: 4 Total number of outcomes: 6 P(A') =
= 6
3
Let us add them: P(A) + P(A') = 1 + 2 = 3 = 1 9/16
3
3
3
That makes 1 It makes sense, right? Event A plus all outcomes that are not Event A make up all possible outcomes. Video Complement of an Event - Probability
Why is the Complement Useful? It is sometimes easier to work out the complement first.
Example. Throw two dice. What is the probability the two scores are different? Different scores are like getting a 2 and 3, or a 6 and 1. It is quite a long list: A = { (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), ... etc ! }
But the complement (which is when the two scores are the same) is only 6 outcomes: A' = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) } And the probability is easy to work out: P(A') = 6/36 = 1/6
Knowing that P(A) and P(A') together make 1, we can calculate: P(A) = 1 - P(A') = 1 - 1/6 = 5/6 So in this case it's easier to work out P(A') first, then find P(A)
Lesson 2
10/16
Conditional Probability How to handle Indipendent and Dependent Events The concept of conditional probability is related to the concept of independent and dependent events.
Independent Events Two events, A and B are independent if the probability of A given B is the probability of A: P(A|B) = P(A). Events can be "Independent", meaning each event is not affected by any other events.
Example: Tossing a coin. Each toss of a coin is a perfect isolated thing. What it did in the past will not affect the current toss. The chance is simply 1-in-2, or 50%, just like ANY toss of the coin. So each toss is an Independent Event.
more……
Dependent Events But events can also be "dependent" ... which means they can be affected by previous events ... When two events are dependent, then the outcome of one event affects the outcome of the other event. In probability notation, the two events B and A are dependent if either P(B|A) ≠ P(B) or P(A|B) ≠ P(A). Two events A and B in a probability space are known to be dependent events, if the occurrence of one affects the occurrence of the other.
11/16
Example: Marbles in a Bag 2 blue and 3 red marbles are in a bag. What are the chances of getting a blue marble? The chance is 2 in 5 But after taking one out you change the chances!
So the next time:
if you got a red marble before, then the chance of a blue marble next is 2 in 4
if you got a blue marble before, then the chance of a blue marble next is 1 in 4
See how the chances change each time? Each event depends on what happened in the previous event, and is called dependent. That is the kind of thing we will be looking at here
Summary:
The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A), read as the probability of B given A. The formula for conditional probability is:
The probability of event B given event A equals the probability of event A and event B divided by the probability of event A The Venn Diagram below illustrates P(A), P(B), and P(A and B). What two sections would have to be divided to find P(B|A
To get the conditional probability, divide the overlap of the two circles by the circle on the left. P(B|A) = P(A and B) divided by P(A Video: How to Calculate Conditional Probability
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1. Example: Ice Cream. 70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberr y. What percent of those who like Chocolate also like Strawberry? Solution: P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate) 0.35 / 0.7 = 50% 50% of your friends who like Chocolate also like Strawberry
2. Example: Tests A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test? Analysis: This problem describes a conditional probability since it asks us to find the probability that the second test was passed given that the first test was passed. In the last lesson, the notation for conditional probability was used in the statement of Multiplication Rule 2. Solution:
P(Second|First)
=
P(First and Second) 0.25 = = 0.60 = 60% P(First) 0.42
worksheets
Comprension
The Normal Distribution (Bell Curve)
In the 1700’ mathematician de Moivre tossing (a) coins and (b)counting the ways he got heads or tails, discovered that his results made patterns. He translated these patterns into (c)“bell curve”. In the early 1800's, German mathematician and physicist Karl Gauss used it to analyze astronomical data, and it consequently became known as the Gaussian distribution among the scientific community In many natural processes, (d) random variation conforms to a particular (e) probability distribution is known as the normal distribution, which is the most commonly observed probability distribution..
13/16
The shape of the normal distribution resembles that of a bell, so it sometimes is referred to as the "bell curve", an example of which follows. Today, the theory of probability is a ( f) fundamental idea in modern life. It is (g )vital in many professions and organisations.
Normal Distribution
Video… Advanced…Normal distribution
Look at the words labelled (a) –(j) and choose the best meaning. a Tossing: throwing / finding b Counting: action of finding the number of elements / to calculate approximately c Bell curve:kind of function whose graph is a bell-shaped curve / a signal that a bar in the corner is closing soon . d Random: arbitrary / casual e Probability distribution: listing of all the values the random variable can assume with their corresponding probabilities / publishing probability. f Fundamental: basic / funny g Vital: important /expensive
Key words
Certain
An event is certain if it will definitely happen
Event
single observation in a probability experiment
Experiment
an action where the result is uncertain.
Impossible
an event is impossible if it can never happen
Likely
probable
Sample Point
just one of the possible outcomes
Sample Space
all the possible outcomes of an experiment 14/16
Trial
A single observation in a probability experiment
Word used in the study of probability Probability wordsearch
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CERTAIN; CHANCE; ESTIMATE; EXPERIMENT; FAIR; IMPOSSIBLE; OUTCOME; PROBABILITY; RELIABLE; SAMPLE; SPINNER; THEORETICAL.
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Sites and bibliography http://www.mathsisfun.com/data/probability.html Mathematics – C. Meyrick, J. Roberts –Oxford (Content and Language support) http://www.mathgoodies.com/lessons/vol6/conditional.html http://www.netmba.com/statistics/distribution/normal/ Complementary events and dependent events How to Calculate Conditional Probability http://www.topperlearning.com/ (Video Lessons Understand / Revise Course)
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