Probability

1. Probability Year 10 IGCSE – Chapter 10

2. What are the chances? What is the probability of winning the first prize in Lotto? Why to people toss a coin to make a decision? What other tools do we use in a game of chance?

3. What are the chances? Say whether each of these events is ‘certain’, ‘likely’, ‘unlikely’ or impossible to occur. a.) You will live in the same house for the rest of your life b.) You will toss a die and roll a ‘six’ c.) The sun will set in the west tonight d.) It will be colder where in February than in August e.) The mail will be delivered tomorrow f.) It will rain next month

4. Probability Tables List all the possible totals you can roll with two normal dice in a table. 4 2 3 5 6 7 3 8 4 5 6 7 What is the probability of obtaining: a.) a total of 5 b.) a total of 11 c.) a ‘two’ on the black die, and a ‘six’ on the white die. 9 4 5 6 7 8 5 9 10 6 8 7 = 10 6 9 11 7 8 7 8 9 12 10 11 = How many possible outcomes? 36

5. Probability Tables List all the possible totals you can roll with two normal dice in a table. 4 2 3 5 6 7 3 8 4 5 6 7 Roll two dice 50 times and record the total of the numbers on the dice in a table. 9 4 5 6 7 8 5 9 10 6 8 7 10 6 9 11 7 8 7 8 9 12 10 11 2 Frequency Tables 11 12

6. Exploring Probability 1.) From your table what is your experimentalprobability of rolling: a.) a six? b.) a one? c.) a total less than 4? d.) an even number? e.) not an even number? f.) What happens when you add your answers from part d and e ? g.) Is 1 a possible outcome?

7. Note 1: Simple Probability If a trial has ‘n’ equally likely outcomes, and a success can occur ‘s’ ways, then the probability of a success is: P(success) = e.g. What is the probability of tossing a ‘heads’ Flipping a coin has 2 equally likely outcomes n = 2 Tossing a head is a success, this can only occur 1 way P(heads) =

8. Note 1: Simple Probability If a trial has ‘n’ equally likely outcomes, and a success can occur ‘s’ ways, then the probability of a success is: P(success) = This scale shows how we can describe the probability of an event 0 0.5 1

9. Note 1: Key Ideas Probabilities can be written as fractions, decimals or percentages. Probabilities are always between 0 and 1. The sample space is a list of all possible outcomes. The probability of all possible outcomes always add to 1. IGCSEEx 6 Pg329-331

10. Starter A jar contains a large number of marbles coloured red, green, yellow, orange and blue. A marble was chosen at random, its colour noted and then replaced. This experiment was carried out 200 times. Here are the results. What is the probability that a randomly selected marble is: a.) orange b.) green c.) not green d.) red or blue = = = =

11. Long run Probability Task Experimental probability from an experiment repeated a large number of times can be useful to make predictions about events. Toss a coin 100 times. Record how many times it lands on heads in a table. After every 10 throws calculate the fraction of heads so far. Convert your proportions (fractions) to decimals. Graph the number of throws vs. the proportion of heads.

12. Your table should look like this……. Graph the number of tosses vs. the proportion of heads What do you notice about the proportion of heads tossed as the number of tosses increases?

13. Note 2: Long Run Probability • Long run proportions can be obtained by repeating the experiment a number of times • there will always be some variation in experiments because chance is involved • probability becomes more accurate as more trials are carried out (closer to theoretical probability)

14. Note 3: Equally Likely Outcomes When outcomes of an event are equally likely, their probabilities are the same. If A is a particular event then: P(A) = P(A) means ‘the probability that A will occur’ The compliment (opposite of A) is all the possible outcomes not in A and is written A’ (not A) P(not A) = 1 – P(A) BETAEx 33.03 Pg946-949

15. Note 4: Exclusive and Independent Events Two events are exclusive if they cannot occur at the same time e.g. Rolling a die and having it be an even number and rolling a ‘3’. e.g. Drawing from a pack of cards a black card and a diamond For exclusive events, A and B P(A or B) = P(A) + P(B)

16. Note 4: Exclusive and Independent Events e.g. A marble is selected from a bag containing 3 red, 2 white, and 5 purple. What is the probability of selecting a red OR a white ball? These are exclusive events P(R or W) = P(R) + P(W) = + = =

17. Note 4: Exclusive and Independent Events Two events are independent if the occurrence of one does not affect the other. e.g. Rolling a die and tossing a coin at the same time. For independent events, A and B P(A and B) = P(A) × P(B) e.g. A fair die and a coin are tossed. What is the probability of obtaining a ‘tails’ and an even number on the die? These are independent events P(Tails and even) = P(tails) × P(even) = x =

18. Note 4: Exclusive and Independent Events Two cards are drawn from a pack of 52, one after the other. The first card is replaced before the second card is drawn. What is the probability that both cards are Aces? P(2 Aces)= P(Ace) x P(Ace) e.g. = x = e.g.The first card is not replaced before the second card is drawn. What is the probability that both cards are Aces? P(2 Aces no replacement) = P(Ace) x P(Ace) = x =

19. Note 4: Exclusive and Independent Events • The probability that it will rain on any day in May is . • Find the probability that: • a.) it will rain on both May the 1st and May the 21st. e.g. = x = • b.) it will not rain on May the 21st. • P(not rain) = 1 – P(rain) = 1 - = • c.) it will rain on May the 1st, but not on May the 21st. • P(rain and not rain) = x = IGCSEEx 7 Pg332

20. Note 5: Predicting Numbers If we know the probability of an event, we can predict roughly how often the event will occur. Expected Number = Number of trials x Probability of event e.g. How many times would we expect a ‘three’ to occur when a fair die is rolled 120 times. P(three) = Number of trials = 120 Expected number of ‘threes’ = 120 x = 20

21. Note 5: Predicting Numbers Expected Number = Number of trials x Probability of event e.g. When playing basketball the probability of getting a basket from inside the key is 0.75. If you make 20 shots, how many can you expect to go in? P(basket) = 0.75 Number of trials = 20 Expected number of ‘baskets’ = 20 x 0.75 = 15

22. Note 5: Predicting Numbers Expected Number = Number of trials x Probability of event e.g.The percentage of students that pass an examination is 45%. If 700 students sit the examination, how many students would be expected to pass? Number of students = 700 x 0.45 = 315 BETAEx 33.04 Pg953

23. Note 6: Tree Diagrams • Tree diagrams are useful for listing outcomes of experiments that have 2 or more successive events • (choices are repeated) • the first event is at the end of the first branch • the second event is at the end of the second branch etc. • the outcomes for the combined events are listed on the right-hand side.

24. Note 6: Tree Diagrams The probability of some events can also be found using a probability tree. Each branch represents a possible outcome. Branch A node is a point where a choice is made. Node

25. Note 6: Tree Diagrams e.g. The possibilities when a couple have 2 children are: B • Every possible outcome must be represented by a branch from a node B G  The sum of the probabilities on the branches from each node is 1 B G  To calculate the probabilities of a sequence of events, we multiply the probabilities along the branches. G e.g. P(B,B) = x =

26. Note 6: Tree Diagrams How many possible results are there? 2 x 3 = 6 R SR H SH Find(random select) P(single) = P(single vanilla)= P(raspberry)= DV V D R DR H DH =

27. Note 6: Tree Diagrams BETAEx 30.05 Pg443-444 Draw a tree diagram to show some alternative ways you could spend your Saturday You must first do a chore (choose from 4 options), Then you can choose to either watch a DVD or visit friends (2 options) 4 x 2 = 8 possible outcomes

28. Note 6: Tree Diagrams A bag contains 5 red balls and 3 green balls. A ball is selected and then replaced. A second ball is selected. Find the probability of selecting: a.) Two green balls R R G x = R G G

29. Note 6: Tree Diagrams A bag contains 5 red balls and 3 green balls. A ball is selected (NOT replaced). A second ball is selected. Find the probability of selecting: a.) Two green balls R R G x = = R G b.) One red and one green G x + x = =

30. Note 6: Tree Diagrams IGCSEEx 8 Pg333-336 On a Monday or a Thursday, Mr. Picasso paints a ‘masterpiece’ with a probability of . On any other day, the probability of producing a masterpiece is . Mr. Picasso never knows what day it is, so what is the probability that on a random day he will produce a masterpiece? There are 7 days in the week there is a probability of there is a probability of x + x =

31. Note 7: Venn Diagrams A venn diagram presents information in groups.

32. Note 7: Venn Diagrams Set A can be written: A = {a, b, c, e} The number of elements in set A is 4, written n(A) = 4 Set B = {c, d, e, f, h} and n(B) = 5 The rectangle ε represents the universal set ε = {a, b, c, d, e, f, g, h} and n(ε) = 8

33. Note 7: Venn Diagrams The overlap of the two set represents the intersection of the sets. A ∩ B = {c, e}n(A ∩ B) = 2 The union is the set of elements in A or B or in both sets. A U B = {a, b, c, d, e, f, h} n(A U B) = 7

34. Note 7: Venn Diagrams A’ is the complement of the set A. It contains all the elements of ε which are not in A. n(A’) = 4 A’ = {d, f, g, h} {g} (A U B)’ = (A ∩ B)’ = A only = {a, b, d, f, h, g} {a, b}

35. Note 7: Venn Diagrams e.g. • In a year 10 class of 25 students, 18 enjoy watching basketball and 15 enjoy watching tennis. If a student is chosen at random, find the probability that he: • a.) enjoys watching both basketball and tennis n(B) = 18 n(T) = 15 n(ε) = 25 n(B ∩ T) = 8 BB Tennis P(B ∩ T) = 10 8 7 • b.) enjoys watching tennis only P( T only) =