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##### Chance and Data

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**Chance and Data**Year 11 1.12**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?**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**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**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) =**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**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. GammaEx 22.01 Pg312**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 = = = =**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.**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?**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)**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) GammaEx 22.02 Pg314**Note 4: Expected Value**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**Note 4: Expected Value**Expected Number = Number of trials × Probability of event Expected Number = n ×p 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**Note 4: 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 NuLake Pg212-213**Note 5: 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’. For exclusive events, A and B P(A or B) = P(A) + P(B)**Note 5: 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) = + = =**Note 5: 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 =**Note 5: 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 =**Note 5: 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 = NuLake Pg215**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.**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**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 =**GammaEx 22.04**Pg323 Note 6: Tree Diagrams 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**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**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 = =**Note 6: Tree Diagrams**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 =**Note 7: Relative Frequency**Results from experiments can be used to predict outcomes of future events. Experiments that are repeated a large number of times have probabilities that approach a consistent value – long run relative frequency**Note 7: Relative Frequency**e.g. A survey was conducted to question people about their smoking habits. 640 360 380 620 1000 a.) How many people were surveyed? b.) Find the probability that a randomly selected person is a smoker 380/1000 = 0.38 c.) Find the probability that a randomly selected person is a female, non-smoker. = 0.21 210/1000**Note 7: Relative Frequency**e.g. A survey was conducted to question people about their smoking habits. 640 360 380 620 d.) Find the probability that a female selected is a non-smoker? 210/360 = 0.58 (2 dp) e.) If 15% of all smokers roll their own cigarettes, how many of the people surveyed would you expect to roll their own? 0.15 x 380 = 57 people**Note 7: Relative Frequency**A six-sided die was rolled 50 times, below are the results. P(1)=0 a.) Find the relative frequency of the number 1. b.) Find the relative freq. of prime numbers. c.) Find the relative freq. of a number less than 3. d.) How would you check if this is a biased die? P(prime)= P( < 3 ) = Conduct a large number of repeated trials.**Note 7: Relative Frequency**A shipment of electronics are inspected for defects. It is found that out of 30 cases, 20% were defected. Find the probability that: a.) the next 3 cases will be defected b.) the next 3 cases will have no defects P(nxt 3 def.) = 1/5 x 1/5 x 1/5 = 1/125 0.008 P(nxt 3 def.) = 4/5 x 4/5 x 4/5 = 64/125 0.512**Starter**A café has recorded its customers’ preferences for tea and coffee, and whether or not they add sugar and milk to their drink. This table gives the results for customers who order a single drink. a.) What is the probability that a customer selected at random has a cup of tea? 131/295 = 44.4% b.) What is the probability that a customer adds both milk and sugar to their drink? 80/295 = 27.1% c.) What is the probability that a customer has a drink with milk added? 185/295 = 62.7% d.) What is the probability that a coffee drinker likes their coffee black with no sugar? 55/164 = 33.5%**Note 8: Calculating Averages**• In statistics, there are 3 types of averages: • mean • median • mode Mode Median Mean - x The middle value when all values are placed in order The most common value (s) Affected by extreme values Not Affected by extreme values These are all measures of central tendency**Note 8: Quartiles**An indication of the spread of data. Lower Quartile – Q1 Upper Quartile – Q3 Median of bottom half Median of top half First identify the median to split the data into halves – do not include the median in either of these halves e.g. 40, 41, 42, 43, 44, 45, 49, 52, 52, 53 LQ median UQ Range– how spread out the data is. It is the difference between the maximum and minimum values Inter-quartile Range - the difference between the UQ & LQ– measuresthe spread of the middle 50% of data**Note 8: Quartiles**e.g. Calculate the median, and lower and upper quartiles for this set of numbers 35 95 29 95 49 82 78 48 14 92 1 82 43 89 Arrange the numbers in order 1 14 29 35 43 48 49 78 82 82 89 92 95 95 LQ UQ median Median – halfway between 49 and 78, i.e. = 63.5 LQ – bottom half has a median of 35 UQ – top half has a median of 89**Note 9: Statistical Tables**Tables are efficient in organising large amounts of data. If data is counted, you can enter directly into the table using tally marks The 33 student in 11JI were asked how many times they bought lunch at the canteen. Below is the tally of individual results. 0 4 0 3 5 0 5 5 0 2 1 0 5 2 3 0 0 5 5 1 2 5 5 3 0 0 1 5 0 5 1 3 0 The data can be summarised in a frequency table**Note 9: Statistical Tables**Calculate the mean = = = = 2.3 Why is this mean misleading? Most students either do not buy their lunch at the canteen or buy it there every day. Total 33**47.In a javelin competition two competitiors were vying to**represent their province in a national competition. On the basis of these results who would you select and why? Results in metres Peter: 42.4, 39.5, 43.2, 47.2, 31.6, 40.2, 41.4, 38.5, 29.5, 34.4 Quade: 37.8, 41.2, 40.8, 42.4, 41.2, 36.7, 42.3, 41.9, 34.2, 35.7 Quade is more consistent – Lower inter-quartile range & stnd dev. 38.8 39.4 29.5 34.2 Peter has a higher UQ and longer maximum throw Which is more important ? 34.4 36.7 39.85 41.0 42.4 41.9 47.2 42.4 Choose: Peter 5.2 2.9 8.0 5.2 NuLake Q47. pg 229**Note 10: Data Display**Box and Whisker Plot – comparing data Male Female x median minimum maximum Upper quartile extreme value Lower quartile IQR**Note 10: Data Display**Line Graphs – identify patterns & trends over time Interpolation - Reading in between tabulated values Extrapolation - Estimating values outside of the range Looking at patterns and trends 0 1 2 3 4 5 6 7 8 9 10 11**Note 10: Data Display**Pie Graph – show proportion Multiply each percentage of the pie by 360° 60% - 0.6 × 360° = 216° 60 Scatter Graph – show relationship between 2 sets of data Plot a number of coordinates for the 2 variables Draw a line of best fit - trend Reveal possible outliers (extreme values)**Note 10: Data Display**Histogram – display grouped continuous data – area represents the frequency frequency Bar Graphs – display discrete data Distance (cm) – counted data – draw bars (lines) with the same width – height is important factor**Note 10: Data Display**Stem & Leaf– Similar to a bar graph but it has the individual numerical data values as part of the display – the data is ordered, this makes it easy to locate median, UQ, LQ Back to Back Stem & Leaf– useful to compare spread & shape of two data sets 5 10 3 3 4 8 9 8 8 3 11 2 3 6 7 8 8 4 2 0 12 1 9 9 3 3 13 0 2 2 14 5 Key: 10 3 means 10.3