Lesson 1 Possibility space diagrams Cambridge IGCSE Mathematics

Lesson 1 – Possibility space diagrams Cambridge IGCSE™ Mathematics 0580 Version 1. 0

Lesson objectives To be able to calculate the probability of two combined events using a possibility space diagram

How is probability relevant? But how do you calculate the probability of matching 6 balls to win a game?

Lets play! We are going to play our own version of the game To play you need to pick four different numbers between 1 and 12 Write them down in your book I am going to use the total score from two dice rolls to pick the numbers To win you need to be the first to cross off all 4 of your numbers. WRITE DOWN YOUR FOUR NUMBERS !!!

Lets play! We are going to play the game again What did you learn from the last game? How can you maximise your chances of winning? Remember to pick four different numbers between 1 and 12 To win you need to be the first to cross off all 4 of your numbers WRITE DOWN YOUR FOUR NUMBERS !!!

Lets play! 1 st Dice Roll 1 2 3 4 5 6 1 2 2 nd Dice Roll 3 4 5 6 We can see what outcomes are most common by drawing a Possibility Space Diagram.

Lets play! 1 st Dice Roll 2 nd Dice Roll 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12 When picking your 4 numbers which are the best to choose? How does the diagram help you to decide?

Try this! G R B B I Spin the spinner twice and I only win if I get the exactly the same colour on both spins. G Draw a Possibility Space Diagram. B R B Use it to calculate the probability that I win.

Try this! Spin 2 R G R B B R R G Spin 1 G B B R G G B B

Try this! Spin 2 G R B B G B B R Spin 1 R R G G B B R RR RR RG RG RB RB G RG RG GG GG GB GB B RB RB BG BG BB BB BB BB

Is rock, paper, scissors a fair game? Player 1 Player 2 D W L L D W W L D Probability of winning, drawing and losing if you are player one is 1/3 - so a fair game! Is this true in practise? Try it!

Is rock paper scissors a fair game? Studies show that In reality things are not quite as simple! Psychological factors come into play based on experience of the players. Inexperience players often avoid making the same selection consecutively as they don’t believe it is random enough, when in reality this would happen randomly 1/3 of the time! Tournament play involving experienced players also show that psychological differences might make things less straight forward. A scientific survey in 2008 found the following selections were made at tournaments: Rock selection 35. 4% Paper selection 36. 0% Scissor selection 29. 6% Scissors is the least popular choice, and men favour rock. Both are reasons to choose paper in a one-shot match!

Try this! 1) I have two bags with 4 pieces of paper in each. In the first bag the pieces are labelled: 1, 2, 3 and 4. In the second bag the pieces are labelled: 3, 4, 4 and 5. I take a piece of paper from each bag and add the scores together. a) Draw a possibility space diagram to show all the possible outcomes. b) What is the probability that the final score is eight? c) What is the probability that the final total is bigger than six? + 1 2 3 3 4 5 6 4 4 6 7 4 5 6 7 8 4 7 8 8 9

Lesson 2 – Finding probabilities using fraction multiplication Cambridge IGCSE™ Mathematics 0580 Version 1. 0

Lesson objectives To be able to calculate probabilities of combined events using fraction arithmetic instead of a possibility space diagram.

Fraction arithmetic - recap 4 3

Try this! A bag contains 5 tickets. 3 of them winning tickets. 2 are not winning tickets. A student takes two tickets from the bag. Before drawing the second ticket the first ticket must be returned to the bag. Draw a possibility space diagram to show the number of different ways that two tickets can be drawn from the bag. Use this to calculate the probability that the student draws two winning tickets.

2 nd ticket Try this! 1 st ticket + W W W x x L L W L L x x x Notice that: P(getting 2 wins) = P(1 st ticket wins) × P(2 nd ticket wins) In general it is true that: P(A followed by B followed by C …. . ) = P(A) × P(B) × P(C) …………

Try this!

Try this! I spin a coin three times. What is the probability that it will show heads, then tails, then heads?

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Try this! A bag contains 5 red balls and 3 green balls. I take a ball out of the bag record its colour and then return it to the bag before drawing a second ball. What is the probability that both balls are red? What is the probability that I get a red ball followed by a green ball?

Try this! A bag contains 5 tickets. 3 of them winning tickets. 2 are not winning tickets. A student takes two tickets from the bag. Before drawing the second ticket the first ticket must be returned to the bag. Lets revisit our ticket problem again: Suppose the student wins if they get just one winning ticket. How can we calculate the probability of them winning?

2 nd ticket Try this! + W W W L L 1 st ticket W W W L x x x x x L x x x There are two ways we can with exactly one ticket: Win the first ticket and lose the second ticket = Lose with first ticket and win with second ticket = To get the answer we need to add these together =

Lesson 3 – Draw and interpret tree diagrams Cambridge IGCSE™ Mathematics 0580 Version 1. 0

Lesson objectives • To be able to draw and interpret probability tree diagrams. • Use probability tree diagrams to solve probability questions involving independent events.

Finding probabilities - recap A bag contains six tickets numbered 1, 2, 3, 4, 5 and 6. A student plays a game where they draw a ticket from the bag. If they get a number less than 3 they eat a chocolate. They play the game twice, returning the ticket between games so that there always six tickets in the bag. Find, using any method, the probability that: a) The student eats two chocolates. b) The student eats no chocolates. c) The student ends up eating exactly one chocolate. d) The student eats at least one chocolate.

Finding probabilities - recap 1 st game 2 nd game + 1 2 3 4 5 6 1 x x 2 x x 3 4 Calculate the probability that: 5 6 a) The student eats two chocolates. b) The student eats no chocolates. c) The student ends up eating exactly one chocolate. d) The student eats at least one chocolate. P(Student eats 2 chocolates) = P(Win Game 1) × P(Win game 2) = = =

Finding probabilities - recap 1 st game 2 nd game + 1 2 3 4 5 6 1 2 3 4 Calculate the probability that: 5 6 a) The student eats two chocolates. b) The student eats no chocolates. x x x x c) The student ends up eating exactly one chocolate. d) The student eats at least one chocolate. P(Student eats no chocolates) = P(Lose Game 1) × P(Lose game 2) = = =

Finding probabilities - recap 1 st game 2 nd game + 1 2 3 4 5 6 1 x x 2 3 x x 4 x x Calculate the probability that: 5 x x 6 x x a) The student eats two chocolates. b) The student eats no chocolates. x x c) The student ends up eating exactly one chocolate. d) The student eats at least one chocolate. P(Student eats 1 chocolate) = P(Win Game 1) × P(Lose Game 2) + P(Lose Game 1) × P(Win Game 2) = =

Finding probabilities - recap 1 st game 2 nd game + 1 2 3 4 5 6 1 x x x 2 x x x 3 x x 4 x x Calculate the probability that: 5 x x 6 x x a) The student eats two chocolates. b) The student eats no chocolates. c) The student ends up eating exactly one chocolate. d) The student eats at least one chocolate. P(Student eats at least 1 chocolate) = P(eat 1) + P(eat 2) = =

Tree diagrams Here is a different way to show the outcomes for my game. 1 st game W Start by considering the 1 st game only. L He can either win and eat the chocolate or lose.

Tree diagrams Here is a different way to show the outcomes for my game If he wins the 1 st game, he still has a second game to play. He might still win or lose the second game. 2 nd game W 1 st game W L L

Tree diagrams If he lost the 1 st game, he still has a second game to play. He might still win or lose the second game. 1 st game W 2 nd game L W L

Tree diagrams There are 4 possible outcomes and to calculate the probability of each outcome you multiply the probabilities on each branch together. 2 nd game Outcomes W 1 st game P(W then W) W L P(W then L) W P(L then W) L L P(L then L)

Tree diagrams We can show the outcome of 2 games as follows 2 nd game Outcomes W 1 st game P(W then W) W L W P(W then L) L P(L then L) L a) The student eats two chocolates. P(L then W)

Tree diagrams We can show the outcome of 2 games as follows 2 nd game Outcomes W 1 st game P(W then W) W L W P(W then L) L P(L then L) L b) The student eats no chocolates. P(L then W)

Tree diagrams We can show the outcome of 2 games as follows 2 nd game Outcomes W 1 st game P(W then W) W L P(W then L) W P(L then W) L L P(L then L) c) The student only eats one chocolate.

Tree diagrams We can show the outcome of 2 games as follows 2 nd game Outcomes W 1 st game P(W then W) W L P(W then L) W P(L then W) L L P(L then L) d) The student eats at least one chocolate

Try this! A bag contains six tickets numbered 1, 2, 3, 4, 5 and 6. A student plays a game where they draw a ticket from the bag. If they get a number less than 3 they eat a chocolate. They play the game twice, returning the ticket between games so that there always six tickets in the bag. What happens if the first ticket is not returned to the bag before the second ticket is drawn? Can I draw a possibility space diagram for this situation? Can I draw a tree diagram for this situation? What is the probability of winning exactly one game now?

Try this! 2 nd game W 1 st game Outcomes P(W then W) W L W P(W then L) P(L then W) L L P(Win exactly one chocolate) = P(L then L)

Lesson 4 – Conditional probability Cambridge IGCSE™ Mathematics 0580 Version 1. 0

Lesson objectives To understand be able to calculate conditional probabilities.

Try this! Well it would be a half if we were not told in the question that “one of the children is a boy”

Try this!

Conditional probability is all about recalculating the odds based on some extra information that you have been given in the question. It is easy to miss what you do truly know.

Conditional probability Consider these two questions separately and think about the differences between them. One is a conditional probability question and one is not. Question 1 I roll a fair six sided die, what is the probability that I roll an even number? Question 2 I roll a fair six sided die, what is the probability that I roll an even number if the number I get is prime? The extra information lets us cut down the number of items in the possibility

Try this! In a class of 25, 12 students have blonde hair and 8 students have glasses. 2 students with glasses also have blonde hair. E a) Draw a Venn diagram for this information. 7 10 Blonde If I select a student at random, what is the probability that: 2 6 Glasses b) They have blonde hair but do not wear glasses? c) Have neither blonde hair or glasses?

Try this! E 7 If I select a student with glasses what is the probability that they have blonde hair? 10 Blonde 2 6 Glasses The student has to be one of these 8, 2 of which have blond hair. So

The higher or lower game In the game you have 6 cards. 1. Each card has a different integer on it from 1 to 6. 2. The first card is revealed – you need to turn over all six cards by correctly ‘guessing’ if the next card will be higher or lower than the current card. 3. If you manage to turn over all 6 cards correctly you win the game. 6 3 2 1 5 4 How do we use conditional probability to assist us in playing this game?

Try this! In a class of 30 students, 10 study biology, 17 study history and 5 study both history and biology. E 8 a) Draw a Venn diagram for this information. 5 5 Biology 12 History b) If I select a history student at random, what is the probability that they also study biology?

Try this! E 30 students are asked whether they do each of three activities: Archery (A), Badminton (B), Chess (C). The results are shown in the Venn diagram. a) How many did only badminton? 8 b) What is the probability that a student chosen at random does chess? 6 A 4 8 B 7 5 C c) What is the probability that a randomly chose student does both chess and badminton? d) I pick a chess playing student, what is the probability that they also play badminton?

Lesson 5 – Tree diagrams and more complex probabilities Cambridge IGCSE™ Mathematics 0580 Version 1. 0

Lesson objectives To be able to use tree diagrams to find the probability of more complex combined events involving conditional probability.

Try this! Four pairs of socks are jumbled up in a drawer. If you put your hand in without looking, how many socks must you take out to be certain of getting a matching pair? What if there were 5 pairs? …. 6? Generalise!

Try this! In a bag, there are 3 milk chocolates (M) and 2 dark chocolates (D). If I pick a sweet from the bag, eat it, and then take a second sweet from the bag, what is the probability that they are different types of chocolate? 2 nd sweet P(M then M) M 1 st sweet P(M then D) M D D P(D then M) P(D then D)

Try this! 2 nd sweet M 1 st sweet M D P(M then M) P(M then D) M P(D then M) D D P(D then D) P(I eat two different types of sweets) =

Try this! Carolyn has 18 biscuits in a tin, 10 plain biscuits, 6 chocolate biscuits and 2 ginger biscuits. She takes two biscuits at random out of the tin. What is the probability that the two biscuits are not the same type? PP PC CP CC CG GP GC GG PG Answer =

Try this! My counter is on square number 1. I spin a fair spinner numbered from 1 to 3 and move forward the number of squares stated. If I land on a grey square I am out. What is the probability that I will be out of the game at some point in the next two goes? 1 2 3 4 5 The only way not to lose is to get a 1 followed by a 3. P (win ) = So P (lose) =