PROBABILITY

1. PROBABILITY

2. Probability Concepts - Probability is used to represent the chance of an event occurring - Probabilities can be represented by fractions, decimals and percentages but NOT BY RATIOS - Probabilities lie between 0 and 1 inclusive - Impossible events have a probability of 0 - Certain events have a probability of 1 - The probability of event A is written as P(A) - To determine probability we need to know the sample space (total number of possible outcomes in an event)

3. Experimental Probability - Also known as long run relative frequency - When experimental results are used to determine the probability - Is calculated using: P(A) = Number of Times Event Occurs Total Number of Trials e.g. A drawing pin is tossed 300 times and lands pin down 204 times. Find the probability a tossed pin will land pin down Number of Times Event Occurs = 204 Total Number of Trials = 300 P(Landing Pin Down) = 204 300 17, 0.68, or 68% also acceptable 25 e.g. A basketball player has 26 shots at the hoop and converts 22 of them. Find the probability the player will miss a shot. Number of Times Event Occurs = 4 (26 – 22) Total Number of Trials = 26 P(Misses shot) = 4 26 2 , 0.15 (2 d.p.), or 15% also acceptable 13

4. Two Way Tables - When tables are used to calculate experimental probabilities. - Always make sure to check which group the person/item is being chosen from. e.g. A class of 35 students recorded the following sports they each played. 18 17 18 8 9 35 Calculate the probability that: a) A student chosen at random is a boy P(Boy) = 18 35 b) If a girl is chosen at random, she plays football P(Football Girl) = 8 17 c) A student chosen at random plays rugby or basketball P(Rugby/BB player) = 17 35

5. Theoretical Probability - When prior knowledge is used to determine the probability - Can only be calculated if events are equally likely to occur - Is calculated using: P(A) = Number of Successful Outcomes Number in Sample Space e.g. Find the probability of getting an even number on a dice. Number of Successful Outcomes = 3 (2, 4, or 6) Number in Sample Space = 6 (1, 2, 3, 4, 5, or 6) P(even number) = 3 6 1, 0.5, or 50% are also acceptable 2 e.g. Find the probability of a nine being drawn from a full deck. Number of Successful Outcomes = 4 (4 suits) Number in Sample Space = 52 (4 suits of Ace to King) P(Nine) = 4 52 1, 0.08 (2 d.p.), or 8% are also acceptable 13

6. 2 or More Events - If the events are independent (have no effect on each other), probabilities can then be multiplied together e.g. Find the probability of getting 3 tails when tossing 3 coins. P(3 tails) = P(tail) × P(tail) × P(tail) = 1 × 1 × 1 2 2 2 = 1 8 0.125 and 12.5% are also acceptable e.g. Find the probability of getting 2 tails and a head when tossing 3 coins. Possible Combinations = TTH, THT, HTT P(2 tails and 1 head) = P(tail) × P(tail) × P(head) × 3 = 1 × 1 × 1 × 3 2 2 2 = 3 8 0.375 and 37.5% are also acceptable

7. Tree Diagrams - Display all outcomes of a probability event - Always multiply along the branches and add the resulting probabilities that match the question e.g. A student walks to school 40% of the time and bikes 60%. If he walks he has an 80% chance of being late, if he bikes it is 10%. Calculate a) P(Student is late) b) P(Student is on time and biking) 1. Set up the tree diagram 2. Add probabilities (decimals/fractions) 3. Determine appropriate branch/branches a) P(Student is late) Late 0.8 = 0.4 × 0.8 + 0.6 × 0.1 Walk 0.4 = 0.38 0.2 On Time • P(Student is on time • and biking) 0.1 Late 0.6 Bike = 0.6 × 0.9 0.9 On Time = 0.54

8. Tree Diagrams Continued - Sometimes harder questions can involve conditional probability e.g. From the previous example, it is found that if a student is late, the probability that they will be given a detention is 0.9. 0.9 Detention Late 0.8 No Detention 0.1 Walk 0.4 0.2 On Time 0.9 Detention 0.1 Late No Detention 0.6 Bike 0.1 0.9 On Time • A student is chosen at random. Calculate the probability that the student biked to school and received a detention for being late P(Student biked and received a detention) = 0.6 × 0.1 × 0.9 = 0.054 b) A student that walks to school is randomly chosen. Calculate the probability that they received a detention for being late. P(Student received a detention given they walked) = 0.8 × 0.9 = 0.72

9. Expected Number - If the probability of an event is known it can be used to predict outcomes or explain events. Expected number = Probability × Number of trials e.g. A coin is tossed 60 times. How many heads would you expect? Expected number = P(Head) × Number of trials = 0.5× 60 = 30 e.g. If 12 cards are randomly selected from a full deck, how many diamonds would you expect? Expected number = P(Diamond) × Number of trials = 0.25× 12 = 3 Note: Expected numbers are only estimations, if you actually carry out the event the results can be quite different and can be used to help decide if the probability tool is biased or not.

10. Probability Simulations - Simulations are used when the actual event cannot be reproduced or it is difficult to calculate the theoretical probability. • A simulation will often either involve the calculation of: • a) The long run relative frequency of an event happening b) The average number of ‘visits’ taken to collect a full set Probability Tools Tools that can be used include: Dice Cards Coins Spinners Random Number Table Random Generator on a Calculator The probability tool chosen must always match the situation appropriately

11. Using the Random Number Generator - The Ran# button on the calculator yields a 3 digit number between 0.000 and 0.999 e.g. To simulate LOTTO balls in NZ use the calculator as follows 40Ran# + 1 (truncating the number to 0 d.p. every time) Number of lotto balls Rounding the numbers up to between 1 and 40 Reading only the whole number (not digits after decimal point) e.g. To simulate a situation where there is a 14% chance of success 100Ran# + 1 (truncating the number to 0 d.p. every time) For success we use the digits 1 – 14 For failure we use the digits 15 - 100

12. Describing Simulations - Use the TTRC method to describe the four most important aspects T - Definition of the probability tool ool - Statement of how the tool models the situation T - Definition of a trial rial - Definition of a successful outcome of the trial (if applicable) R - Statement of how the results will be tabulated giving an example of both a successful and unsuccessful outcome esults - Statement of how many trials should be carried out C alculation - Statement of how the calculation needed for the conclusion will be done Either: Long-run relative frequency = Number of ‘successful’ results Number of trials Or: Mean = Sum of trial results Number of trials

13. Simulation Example 1 What is the probability that a four child family will contain exactly 2 boys and 2 girls? Design a simulation to solve the problem. - A calculator using 10Ran# + 1 where 1 – 5 = boy Tool 6 – 10 = girl - One trial will consist of 4 randomly generated numbers Trial to simulate a 4 child family - A successful outcome would be 2 numbers between 1 – 5 (boys) and 2 numbers between 6 – 10 (girls) Results - Results will be recorded on a table as follows: - 30 trials would be sufficient Calculation - To calculate the estimated probability use: P(2 boys & 2 girls) = number of ‘Successful’ results number of trials

14. Simulation Example 2 e.g. A cereal manufacturer includes a gift coupon in each box of a certain brand. These coupons can be exchanged for a gift when a complete set of six coupons has been collected. What is the expected number of boxes of cereal you will have to buy before you obtain a complete set of six coupons?Design a simulation. Tool - A calculator using 6Ran# + 1 where 1 – 6 = each of the coupons Trial - One trial will consist of randomly generated numbers until all 6 numbers are obtained (all coupons) Results - Results will be recorded on a table as follows: - 30 trials would be sufficient Calculation - To calculate the estimated number of boxes use: Average number = Sum of Number of Boxes number of trials