PROBABILITY Probability Concepts Theoretical Probability vs Relative Frequency Calculating Probabilities Venn Diagrams Intersection of Sets Union of Sets Mutually Exclusive Events Inclusive Events Complementary Events Probability Games in Life

PROBABILITY CONCEPTS Example: Tossing a coin twice Probability The likelihood or chance of an event happening E.g. Heads (50%) or Tails (50%) Outcomes The possible results of an experiment E.g. Heads; Tails / Heads; Heads / Tails ; Heads / Tails; Tails Sample Space The set of all possible outcomes: S={HH,HT,TH,TT} The number of outcomes: (S) = 4 The Coin Toss

Types of Events a) Certain events These events will always happen. E.g. If you toss a coin twice and the event is E = {getting a head or a tail} then you are certain to get a head or tail on each toss. (b) Even chance events 50% probability of an event happening E.g. Consider the experiment with S = {HH, HT, TH, TT} and the event A = {getting two heads or two tails) then event A has 50% chance of happening = {HH, TT}

c) Equally likely events Each event has an equally likely chance of happening. Events are unbiased E.g. If you toss a coin, there is 50% change of getting a Head and 50% change of a Tail (d) Impossible Events The event cannot happen E.g. G = {getting the number 7 from a die that is thrown}. This is impossible because the die doesn't contain the number 7. The chance of this happening is= 0. G= or G = { }, which is called the empty set.

We write probabilities as fractions, decimals or percentages. The probability scale We write probabilities as fractions, decimals or percentages. The less likely an event is to happen, the smaller the fraction. The more likely the probability, the greater the fraction.

THEORETICAL PROBABILITY VS RELATIVE FREQUENCY When all outcomes of an activity are equally likely, you can calculate the probability of an event happening by using the following definition: P(E) = number of favorable outcomes = n (E) total number of possible outcomes = n (S) Probability of throwing a dice and getting a 4 is P(4) = 1 6 The Probability of an Event Marble Example

Basic Probability Example Relative Frequency This method involves many trials. E.g. Suppose you threw a die 200 times and then counted the number of times you got the number 4. You might land up getting this number 33 times. You would then say that the relative frequency of getting the number 4 is = = 14.5%. Only after many trials will the relative frequency get closer to the theoretical probability of . = 16.67%. The Spinner Basic Probability Example

Calculating Probabilities Example 1 Suppose that you are required to draw cards numbered from 1 to 12 out of a hat. The sample space is S = {1,2,3,4,5,6,7,8,9,10,11,12} Consider the following events: a) Event A = {drawing prime numbers} = {2,3,5,7,11} b) Event B= {drawing odd numbers} ={1,3,5,7,9,11}

Example 2 The sample space is S = {1,2,3,4,5,6,7,8,9,10,11,12} c) Event C = {drawing factors of 6} = {1,2,3,6} d) Event D = {drawing numbers greater than 12} = { } e) Event E= {drawing natural number} = {1,2,3,4,5,6,7,8,9,10,11,12}

Example 3 Consider the word MATHEMATICS. A letter is chosen from this word. (a) Find the probability that the letter chosen is M. There are a total of 11 letters in the word. M appears twice. Therefore the probability of getting M is . (b) Find the probability that the letter chosen is not M. The probability is (c) Find the probability that the letter chosen is a vowel {A,E,I,0,U}.

EXERCISE 1. 300 tickets were sold for a raffle. Jason bought 12 tickets. What is the probability that he: (a) will win the prize? (b) will not win the prize? 2. There are 60 boys and 40 girls auditioning for POP IDOLS. The names of the contestants are put into a hat and a name is drawn. What is the probability that: (a) a boy will be selected on the first draw? (b) a girl will be selected on the first draw?

3. A bag contains 6 blue marbles, 5 red marbles, 8 green marbles and 9 white marbles. What is the probability of: (a) drawing a white marble? (b) drawing a green marble? (c) drawing a blue marble? (d) drawing a red marble? (e) drawing a red or blue marble? (f) drawing a blue or green marble? (g) drawing a pink marble? (h) drawing a white, green or red marble? (i) not drawing a blue marble?

4. A card is drawn from a pack of 52 cards. Determine the probability of drawing: (a) a heart. (b) a jack of clubs (c) an ace. (d) a king or queen (e) neither a heart or a spade. 5. A six-sided die is thrown. Determine the probability of: (a) throwing a 6. (b) throwing a 2 or a 4. (c) throwing an odd number. (d) not throwing a 5. Probability Quiz

Venn diagrams Venn diagrams represent the sample space. Example Consider the hat experiment where S = {1,2,3,4,5,6,7,8,9,10,11,12}. Suppose that there are two events: A = {drawing numbers less than or equal to 6} = {1,2,3,4,5,6} B = {drawing numbers greater than 6} = {7,8,9,10,11,12} Represent this on a Venn Diagram: Basic Venn Diagrams

Intersection of sets The intersection occurs where the elements share a common space. These are called inclusive events. Example Consider the hat experiment where S = {l, 2,3,4,5,6,7,8,9,10,11,12}. There are 2 events: C = {drawing a factor of 6} = {1,2,3,6} D = {drawing a factor of 9} = {l, 3,9} Find the elements that intersect. C D={l,3} C and D ={1,3}

Union of sets The union of A and B is an event consisting of all outcomes that are in A or B. Example Determine the union of C and D. C D ={l,2,3,6,9} C or D ={1,2,3,6,9} Here the numbers 4,5,7,8,10,11,12 are excluded from the union of C and D. The number 1 and 3 appear in both set C and D and are written only once in the union set.

Mutually exclusive events Events with no elements in common. Event A and B exclude each other. If A happens, then B cannot happen. Both cannot happen at the same time. Example a) Find the intersection of A and B: A B = { } A and B = { } empty set P(A B) = 0 b) Find the union of A and B: A B = {1;2;3;4;5;6;7;8;9;10;11;12} A or B = {1;2;3;4;5;6;7;8;9;10;11;12} P(A B) = P(A) + P(B) – P( A B)

Practicing Venn Diagrams inclusive events Events with elements in common. Example a) Find the intersection of C and D: C D = {1;3} C and D = {1;3 } b) Find the union of C and D: C D = {1;2;3;6;9} C or D = {1;2;3;6;9} P(C D) = P(A) + P(B) - P(A B) Practicing Venn Diagrams

Playing Cards & Venn Diagrams Complementary events If Event A and Event B is mutually exclusive, then Event A and Event B are complementary. Example a) Find the complement of A. A = {1;2;3;4;5;6} Complement of A = Not A (A’) = B = {7;8;9;10;11;12} P(A) + P(A')= 1 b) Find the complement of B. B = {7;8;9;10;11;12} Complement of B = Not B (B’) = A = {1;2;3;4;5;6} P(B) + P(B')= 1 …. P (not B) = 1 - P(B) Complex Venn Diagrams Playing Cards & Venn Diagrams

EXERCISE Cards numbered from 1 to 12 are put into a box and shaken. Cards are then drawn and replaced. The following events are given: A = {drawing an even number} B = {drawing an odd number} C = {drawing a number greater than 7} D = {drawing a number less than 5} E = {drawing natural numbers less than 7} F = {drawing natural numbers greater than 4}

(a) Draw a Venn Diagram to represent events A and B. (b) Determine P(A B) (c) Determine P(A B) (d) Show that A and B are mutually exclusive. (e) Are events A and B complementary? (f) Draw a Venn Diagram to represent events A and C. (g) Determine P(A or C) (h) Determine P(A and C) (i) Show that A and C are inclusive. (j) Are events A and C complementary? (k) Draw a Venn Diagram to represent events C and D. (1) Determine whether C and D are mutually exclusive or inclusive; complementary or not complementary. (m) Draw a Venn Diagram to represent events E and F. (n) Determine whether E and F are mutually exclusive or

PROBABILITY GAMES IN LIFE Picking Cards or Rolling Die Conditional Probability: Pick the Correct Door!