PROBABILITY CONTENTS BASIC PROBABILITY INTRODUCTION BASIC PROBABILITY EXAMPLEPROBLEM
PROBABILITY | CONTENTS BASIC PROBABILITY INTRODUCTION BASIC PROBABILITY EXAMPLE-PROBLEM PAIRS 28/12/2021 PROBABILITY DISTRIBUTIONS INTRODUCTION PROBABILITY DISTRIBUTIONS EXAMPLE-PROBLEM PAIRS MUTUALLY EXCLUSIVE EVENTS PROBABILITY DISTRIBUTIONS EXERCISE INDEPENDENT EVENTS BINOMIAL DISTRIBUTIONS INTRODUCTION COMBINED EVENTS EXAMPLE-PROBLEM PAIRS BASIC PROBABILITY EXERCISE 1
MINI WHITEBOARD QUESTION 28/12/2021 Two fair spinners each have four sectors numbered 1 to 4. The two spinners are spun together and the sum of the numbers indicated on each spinner is recorded. Find the probability of the spinners indicating a sum of (a) exactly 5 (b) more than 5 Spinner 2 Spinner 1 + 1 2 3 4 5 6 3 4 5 5 6 6 7 7 8 4 IF THE SAMPLE SPACE IS A COMBINATION OF TWO UNDERLYING EXPERIMENTS, A TABLE IS A HELPFUL WAY TO LIST THE OUTCOMES. 2
PROBABILITY INTRODUCTION 28/12/2021 3 AN EXPERIMENT IS A REPEATABLE PROCESS THAT GIVES RISE A NUMBER OF OUTCOMES. AN EVENT IS A SET OF ONE OR MORE OF THESE OUTCOMES. (WE OFTEN USE CAPITAL LETTERS TO REPRESENT THEM) 1 4 5 2 A SAMPLE SPACE IS THE SET OF ALL POSSIBLE OUTCOMES. 3 6 Because we are dealing with sets, we can use a Venn diagram, where • the numbers are the individual outcomes, • the sample space is a rectangle and • the events are sets, each a subset of the sample space.
SET NOTATION WITH VENN DIAGRAMS 28/12/2021 4
THE COMPLEMENT OF AN EVENT Since the probabilities of all possible outcomes add up to 1, QUICK EXAMPLE Probability of not getting a square number is 28/12/2021 5
BASIC PROBABILITY | EXAMPLE-PROBLEM PAIR 28/12/2021 A card is picked at random from a pack of cards. Find the probability of the events below. Use set notation for your answers. (a) A club is picked (b) A queen is picked (c) The queen of clubs is picked. (d) A club or a queen is picked. (e) A club is not picked. 6
BASIC PROBABILITY | EXAMPLE-PROBLEM PAIR 28/12/2021 A card is picked at random from a pack of cards. Find the probability of the events below. Use set notation for your answers. (a) A picture card (Jack/Queen/King) is picked (b) A heart is picked Reveal Answer (c) The card is both a heart and a picture card. (d) The card is not a picture card. Reveal Answer (e) The card is not a heart and not a picture card. Reveal Answer 7
PROBABILITY | THE ADDITION PROPERTY We subtract the final term as otherwise it would be included twice 28/12/2021 8
MUTUALLY EXCLUSIVE EVENTS 28/12/2021 If two events are mutually exclusive they can’t happen at the same time. The Venn Diagram would look like: 9
MUTUALLY EXCLUSIVE EVENTS | EXAMPLE-PROBLEM PAIR 28/12/2021 A and B are mutually exclusive (common sense) A and B are not mutually exclusive. Draw a Venn Diagram to illustrate your answer. (b) A: Rolling an even number. B: Rolling a prime number. A and B are not mutually exclusive. 10
INDEPENDENT EVENTS 28/12/2021 Two events are independent if whether one event happens does not affect the probability of the other event happening. TOP TIP: INDEPENDENCE DOES NOT AFFECT HOW THE CIRCLES INTERACT IN A VENN DIAGRAM. 11
INDEPENDENT EVENTS | EXAMPLE-PROBLEM PAIR 28/12/2021 12
INDEPENDENT EVENTS | EXAMPLE-PROBLEM PAIR (b) Show that the events are not independent. 28/12/2021 13
COMBINED EVENTS | EXAMPLE-PROBLEM PAIR 28/12/2021 4 P. There are 3 yellow and 2 green counters in a bag. I take two counters at random without replacement. Determine the probability that: (a) They are of the same colour. (b) They are of different colours (a) (b) 1 st counter 2 nd counter 14
COMBINED EVENTS | EXAMPLE-PROBLEM PAIR 28/12/2021 4 E. A bag contains 6 green and 4 blue counters. A counter is chosen at random from the bag three times without replacement. Find the probability that the three counters chosen are: (a) All green 1 st counter 2 nd counter (b) Not all the same colour. (a) (b) 15
PROBABILITY | EXERCISE 21 A MAIN QUESTIONS Q 3 -11 28/12/2021 QUESTIONS: P 468 ANSWERS: P 584 16
PROBABILITY DISTRIBUTIONS 28/12/2021 The list of all possible outcomes together with their probabilities is called a probability distribution. A random variable is a variable that depends on chance The total of all the probabilities in a probability distribution must add up to 1. 17
PROBABILITY DISTRIBUTIONS | EXAMPLE-PROBLEM PAIR + 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12 28/12/2021 - 1 3 5 7 1 0 2 4 6 2 1 1 3 5 3 2 0 2 4 18
PROBABILITY DISTRIBUTIONS | EXAMPLE-PROBLEM PAIR 28/12/2021 19
PROBABILITY | EXERCISE 21 B 28/12/2021 OPTIONAL STARTER QUESTIONS Q 2 (a) (ii) Q 2 (b) (i) MAIN QUESTIONS Q 3 -12 (except Q 5) QUESTIONS: P 471 -473 ANSWERS: P 584 -585 20
INTRODUCTION TO THE BINOMIAL DISTRIBUTION The probability of a person being left handed is 0. 1. Out of 4 people, find the probability of (a) None of them are left handed (b) One people being left handed. (c) Two people being left handed. (d) Three people being left handed. (e) All four are left handed. (a) 28/12/2021 21
INTRODUCTION TO THE BINOMIAL DISTRIBUTION The probability of a person being left handed is 0. 1. Out of 4 people, find the probability of (b) One people being left handed. (b) Here there are 4 possibilities: However, the calculation to work each of these out is the same: (c) Two people being left handed. Here there are 6 possibilities: Again, the calculation to work each of these out is the same: 28/12/2021 22
INTRODUCTION TO THE BINOMIAL DISTRIBUTION The probability of a person being left handed is 0. 1. Out of 4 people, find the probability of (d) Three people being left handed. (d) Here there are 4 possibilities: As before, the calculation to work each of these out is the same: (e) All four are left handed. 28/12/2021 23
INTRODUCTION TO THE BINOMIAL DISTRIBUTION 28/12/2021 A summary: THE RED NUMBERS ABOVE ARE THE BINOMIAL COEFFICIENTS. EXTENSION: IMAGINE THERE ARE 9 PEOPLE. WHAT IS THE PROBABILITY THAT 3 ARE LEFT HANDED? 24
THE BINOMIAL DISTRIBUTION 28/12/2021 The binomial distribution models the number of successful outcomes of n trials, provided that these conditions are satisfied: Each trial has two possible outcomes. The trials are independent of each other. The probability of success is the same in each trial. A fixed number of trials. 25
28/12/2021 26 Which of these situations might reasonably be modelled by a binomial distribution? (a) REVEAL ANSWER (b) (c) Not binomial. The events REVEAL ANSWER are not independent. REVEAL ANSWER
THE BINOMIAL DISTRIBUTION e. g. 28/12/2021 PROBABILITY OF SUCCESS. 50 TRIALS A BINOMIAL DISTRIBUTION PROBABILITY OF 9 SUCCESSES. 27
BINOMIAL DISTRIBUTION | EXAMPLE-PROBLEM PAIRS Extension A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded. (a) Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. (2) 28/12/2021 28
THE BINOMIAL DISTRIBUTION | MINI WHITEBOARDS 28/12/2021 29
BINOMIAL DISTRIBUTION | USING YOUR CALCULATOR 28/12/2021 Although you need to know and be able to use the formula, in most cases you should do these calculations on your calculator. Follow these steps to access binomial distributions: CHOOSE ‘STATISTICS’ (PRESS 2 FROM THE MAIN MENU) CHOOSE ‘DIST’ (DISTRIBUTION F 5) CHOOSE ‘BINOMIAL’ (DISTRIBUTION F 5) BINOMIAL PROBABILITY DISTRIBUTION CUMULATIVE BINOMIAL DISTRIBUTION 30
BINOMIAL DISTRIBUTION | EXAMPLE-PROBLEM PAIRS (a) Find: BINOMIAL PROBABILITY DISTRIBUTION (BPD) (b) Find: CUMULATIVE BINOMIAL DISTRIBUTION (BCD) 28/12/2021 31
PROBABILITY | EXERCISE 21 C 28/12/2021 MAIN QUESTIONS Q 1 (on mini whiteboard if you like) Q 2 (a) (ii), (b)(i) (c) (ii), (d)(i) (f) (i), (f)(ii) Q 3 Q 5 -12 QUESTIONS: P 479 ANSWERS: P 585 32
SKETCHING PROBABILITY DISTRIBUTIONS | EXAMPLE-PROBLEM PAIR 28/12/2021 (b) Sketch the distribution. (a) The event of her staying late each day must be independent. (b) • • • ENTER TABLE MODE (7) ENTER THE BINOMIAL FORMULA AS SHOWN CHOOSE ‘SET’ AND ENTER THE FOLLOWING: • START: 0, END: 6, STEP: 1 • PRESS ‘TABLE’ AND WRITE DOWN THE VALUES. 33
SKETCHING PROBABILITY DISTRIBUTIONS | EXAMPLE-PROBLEM PAIR (b) Sketch the distribution. 28/12/2021 34
SKETCHING PROBABILITY DISTRIBUTIONS | EXAMPLE-PROBLEM PAIR 28/12/2021 Example: The probability that a baby is born a boy is 0. 51. A mid-wife delivers 10 babies. (a) Find: i) the probability that exactly 4 are male; ii) the probability that at least 8 are male. b) Sketch the distribution. (a) (b) 35
SKETCHING PROBABILITY DISTRIBUTIONS | EXAMPLE-PROBLEM PAIR Binomial distribution (c) 28/12/2021 36
AQA S 1 JAN 11 Q 4 28/12/2021 37
AQA S 1 JAN 11 Q 4 28/12/2021 38
AQA S 1 JAN 11 Q 4 28/12/2021 39
AQA S 1 JUNE 2010 Q 4 28/12/2021 40
AQA S 1 JUNE 2010 Q 4 28/12/2021 41
AQA S 1 JUNE 2010 Q 4 28/12/2021 42
AQA S 1 JUNE 11 Q 6 28/12/2021 43
AQA S 1 JUNE 11 Q 6 28/12/2021 44
AQA S 1 JUNE 11 Q 6 28/12/2021 45
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