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Probability is a measure, like length or area or weight or height, but a measure of the likeliness or chance of possibilities in some situation. We can either assume directly that the thrower is equally likely to get any of those places, or we can think of it like the spinner. Probability is a measure, like length or area or weight or height, but a measure of the likeliness or chance of possibilities in some situation. So the list of the possible outcomes of a single spin of the arrow is orange, red, green, blue. A fair die is one for which it is assumed that the 6 faces are equally likely to be uppermost in a toss of the die. Situations involving uncertainty or randomness include probability in their models, and analysis of models often leads to data investigations to estimate parts of the model, to check the suitability of the model, to adjust or change the model, and to use the model for predictions. Many people living in towns or cities arrange for their newspaper to be delivered. Example E: Where does your newspaper land. Example A: Throwing two dice. So we take all the probability and parcel it out amongst the various possibilities. Example C: What colour sweet did you get. Even the most basic exploration and informal analysis involves at least some modelling of the data, and models for data are based on probability. Statistical methods for analysing data are used to evaluate information in situations involving variation and uncertainty, and probability plays a key role in that process. We can model them by considerations of the situation, using information, making assumptions and using probability rules. For some excellent comments on this, see the letter by Harvey Goldstein to the Editor of Teaching Statistics 2010, Volume 32, number 3. Under these carefully described circumstances and assumptions, the probability of each of the list of outcomes is 1/total number of outcomes in the list. Like length or area or weight or height, probability cannot be below 0 − you can’t have less than no chance of something happening. Can probability be 0. In Year 5, consideration of the possible outcomes and of the circumstances of simple situations has lead to careful description of the outcomes and assumptions that permit assigning equal probabilities, understanding what the values of these probabilities must be and representing these values using fractions. Can probability be 0. Almost always we use a combination of assumptions, modelling, data and probability rules. An extension to events used for some games of chance. This situation is far more difficult than is usually assumed in such research which often tends to take the assumptions of fair coins and independent tosses as absolutely immutable and not to be questioned. We have planted 10 seeds in a planter box at the same time, and carefully watered them each day. Probability is a relative measure; it is a measure of chance relative to the other possibilities of the situation. 1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 4, 4, 1, 1, 5, 5, 1, 1, 6, 6, 1, 2, 2, 2, 3, 3, 2, 2, 4, 4, 2, 2, 5, 5, 2, 2, 6, 6, 2, 3, 3, 3, 4, 4, 3, 3, 5, 5, 3, 3, 6, 6, 3, 4, 4, 4, 5, 5, 4, 4, 6, 6, 4, 5, 5, 5, 6, 6, 5, 6, 6.

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Probability is a measure, like length or area or weight or height, but a measure of the likeliness or chance of possibilities in some situation. Example C: What colour sweet will you get. So probabilities for the possibilities of a situation are parts of the whole probability of 1. Where do the values of probabilities come from. Simple everyday events. From Years 1 4, students have gradually developed understanding and familiarity with simple and familiar events involving chance, including possible outcomes and whether they are “likely”, “unlikely” with some being “certain” or “impossible”. We have planted 10 seeds at regular places perhaps using a tray with 10 separate small compartments in a planter box at the same time, and carefully watered them each day. In the spinner pictured below, there are 4 colours, with the list of the possible outcomes of a single spin of the arrow is orange, red, green, blue. Of course there must be at least one of each of the colours in the firework, but do we need to have equal number of the colours in the firework. Example D: Which colour firework goes the highest. What would we need to assume for these 5 areas to be equally likely places for the newspaper to land. Yes, if something is not possible at all, there is no chance it will happen. Sometimes it is very easy to describe possible events and sometimes there is really only one way of describing them, but in many situations this is not so and careful description is therefore important. We can estimate them from data. A list of the possible landing places for the newspaper is footpath but not the driveway, driveway on the footpath, driveway inside the boundary, grass inside the boundary, garden inside the boundary. In some games that use a spinner, if the arrow appears to fall on a line, the spin is taken again. Someone tells us that the possible colours for the stars of a Roman candle are red, blue, green, and yellow and we spread our total probability of one over those four colours in guessing which colour is going to shoot highest.

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The Improving Mathematics Education in Schools TIMES Project 2009 2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The usual single die has 6 sides, with each side having a number of marks giving its face value of one of the values 1, 2, 3, 4, 5 or 6. So we take all the probability and parcel it out amongst the various possibilities. Events used for some games of chance. So the whole probability for a particular situation is 1, and we divide this whole probability into probabilities that are smaller than one and share them around over the various possibilities. All statistical models of real data and real situations are based on probability models. If we assume that the person throwing the newspaper is good enough not to miss our place completely and to avoid it landing in the gutter or on the road, then the list of the possible landing places for the newspaper is footpath but not the driveway, driveway on the footpath, driveway inside the boundary, grass inside the boundary, garden inside the boundary. Commentators often like to ask crowds watching to guess which colour star will be next or which colour star will go the highest. The Improving Mathematics Education in Schools TIMES Project 2009 2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. We will consider situations in which the description of the events is straightforward and natural and easily listed. Someone tells us that the possible colours for the stars of a Roman candle are red, blue, green, and yellow and we spread our total probability of one over those four colours in guessing which colour is going to shoot highest. Someone tells us that the possible colours for the stars of a Roman candle are red, blue, green, and yellow and we spread our total probability of one over those four colours in guessing which colour is going to shoot highest. Equally likely outcomes whose probabilities can be represented as fractions. The basic events are pairs of numbers, where each number is the face value of the uppermost face of one of the dice. But if there are no blue sweets at all in the box, then there’s no chance of a blue − it’s impossible to get a blue if there are no blues in the box. Example E: Where does your newspaper land. In many situations there may be slightly different ways of describing events, but in some simple and everyday situations there is an obvious space xy game and natural way of describing them. A fair die is one for which it is assumed that the 6 faces are equally likely to be uppermost in a toss of the die. What do we need to know or assume, to be able to say that the highest star is equally likely to be any of those 5 colours. Perhaps we should include these possibilities even if they have only a slight chance of happening. In Year 4, they have considered more carefully how to describe possible outcomes of simple situations involving games of chance or familiar everyday outcomes, and, without assigning any values for probabilities, how the probabilities of possible outcomes could compare with each other. The objectives of the chance and probability strand of the F 10 curriculum are to provide a practical framework for experiential learning in foundational concepts of probability for life, for exploring and interpreting data, and for underpinning later developments in statistical thinking and methods, including models for probability and data. Commentators often like to ask crowds watching to guess which colour star will be next or which colour star will go the highest. We can estimate them from data.

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If the dice are fair and they are tossed at random and there is nothing to connect the tosses of each, then there is no reason to assume that any pair of faces is more likely than any other pair. Provided there are some blue sweets in the box, there is some chance a blue sweet will be shaken out. Situations involving uncertainty or randomness include probability in their models, and analysis of models often leads to data investigations to estimate parts of the model, to check the suitability of the model, to adjust or change the model, and to use the model for predictions. In the list of the possible landing places for the newspaper of footpath but not the driveway, driveway on the footpath, driveway inside the boundary, grass inside the boundary, garden inside the boundary, have we considered all possibilities. However, it is more difficult in practice to toss a coin completely randomly than to toss dice randomly, and also to assume that a particular person does not tend to toss a coin in a similar manner each toss. Some get a bigger parcel of probability and some get a smaller parcel. Many people living in towns or cities arrange for their newspaper to be delivered. So the outcomes of a throw or toss of a single die are very simple to describe and there is really no other way of describing the outcome except as the face value of the uppermost face when the die lands, and hence the list of possible events is simply the set of numbers 1, 2, 3, 4, 5, 6. It is assumed that in Years 1 4, students have had many learning experiences that consider simple and familiar events involving chance, including describing possible everyday events and whether they are “likely”, “unlikely” with some being “certain” or “impossible”. We are assuming that the person throwing the newspaper will not miss our place altogether nor will they throw the paper in the gutter or on the road. The list of possible outcomes is the list of pairs of numbers from 1 to 6, where the first number in each pair is the uppermost face of die 1 for example, this might be a red die and the second number in each pair is the uppermost face of die 2 for example, this might be a blue die as follows. If the only possible colours for the stars of a Roman candle are red, blue, green, yellow, silver, then this list is the list of possible outcomes for the colour of the star that goes the highest. Statistical methods for analysing data are used to evaluate information in situations involving variation and uncertainty, and probability plays a key role in that process. The values of probabilities tell us the chance of observing each outcome each time we take an observation, but what can we expect to see in a set of observations. Note that there is a difference between observing a simulation and observing a real situation because assumptions are easier to accept in a simulation. If we assume that the person throwing the newspaper is good enough not to miss our place completely and to avoid it landing in the gutter or on the road, then the list of the possible landing places for the newspaper is footpath but not the driveway, driveway on the footpath, driveway inside the boundary, grass inside the boundary, garden inside the boundary. Therefore each bit of probability is. Assumed background from 1 4. The list of possible outcomes is the list of pairs of numbers from 1 to 6, where the first number in each pair is the uppermost face of die 1 for example, this might be a red die and the second number in each pair is the uppermost face of die 2 for example, this might be a blue die as follows. In many games, such as board games, moves are decided by the chance throws of dice or spins of spinners. The usual single die has 6 sides, with each side having a number of marks giving its face value of one of the values 1, 2, 3, 4, 5 or 6. Statistics and Probability : Module 10Year : 5. Many board games use throws of two dice.

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We can either assume directly that the thrower is equally likely to get any of those places, or we can think of it like the spinner. For either of these procedures, there are 4 possible outcomes of a spin, namely, orange, red, green, blue. If we assume a die is fair, we have 6 possible outcomes all with the same chance of occurring. Then for a finite number of equally likely outcomes of a simple and everyday situation, we can assign the equal probabilities as fractions. In many situations there may be slightly different ways of describing events, but in some simple and everyday situations there is an obvious and natural way of describing them. Almost always we use a combination of assumptions, modelling, data and probability rules. First we need to decide what to do if the arrow appears to land on one of the dividing lines. We are assuming that the person throwing the newspaper will not miss our place altogether nor will they throw the paper in the gutter or on the road. But is this sensible. Equally likely outcomes whose probabilities can be represented as fractions. Probabilities range from 0 to 1. Statistical methods for analysing data are used to evaluate information in situations involving variation and uncertainty, and probability plays a key role in that process.

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We are saying that the thrower won’t miss those 5 places, but the throw is random within that restriction. This work is licensed under the Creative Commons Attribution NonCommercial NoDerivs 3. Statistical methods for analysing data are used to evaluate information in situations involving variation and uncertainty, and probability plays a key role in that process. Example A: Throwing one die. So the whole probability for a particular situation is 1, and we divide this whole probability into probabilities that are smaller than one and share them around over the various possibilities. We now consider assigning probabilities when we have a fixed number of possible outcomes for a simple everyday situation and could consider under certain conditions that all our possible outcomes are equally likely. If the possible colours for the stars of a Roman candle are red, blue, green, yellow, silver, then this list is the list of possible outcomes for the colour of the star that goes the highest. In giving a sweet to your friend by shaking one out of your box of sweets onto your friend’s hand, suppose you have put sweets from a larger container into your small box, and you chose to put in 8 red, 8 yellow, 8 brown, 8 green and 8 blue and shaken the box well. In Year 5, consideration of the possible outcomes and of the circumstances of simple situations has lead to careful description of the outcomes and assumptions that permit assigning equal probabilities, understanding what the values of these probabilities must be and representing these values using fractions. So probabilities for the possibilities of a situation are parts of the whole probability of 1. Equally likely outcomes whose probabilities can be represented as fractions. Probabilities are fractions of the whole probability of 1. For some excellent comments on this, see the letter by Harvey Goldstein to the Editor of Teaching Statistics 2010, Volume 32, number 3. Example D: Which colour firework goes the highest. Example C: What colour sweet did you get. Example F: Which seedling will appear first. Some fireworks used in public fireworks displays send up coloured stars or flashes. Example C: What colour sweet will you get. They have also considered simple everyday events that cannot happen together and, in comparison, some that can. Each time we observe the situation, only one possible outcome can be observed. Therefore each bit of probability is. The probability of a blue one is zero. For example, the list of possible colours might be red, yellow, green, blue, orange, brown. Of course there must be at least one of each of the colours in the firework, but do we need to have equal number of the colours in the firework. So we take all the probability and parcel it out amongst the various possibilities. Usually the different segments are in different colours, so the basic events of a single spin are the colours used for the different segments.

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There is nothing wrong with saying that this is what we are assuming, as long as we do say it clearly. The concepts and tools of probability pervade analysis of data. Example D: Which colour firework goes the highest. Its chance or probability of happening is 0. For some excellent comments on this, see the letter by Harvey Goldstein to the Editor of Teaching Statistics 2010, Volume 32, number 3. Which seed will shoot first − that is, which of the 10 places in which we planted a seed, will we see the first showing of green. Simple everyday events. PDF Version of module. With regard to the assumption of a fair coin and independent tosses, the assumptions are the same as tossing dice. Probability models are at the heart of statistical inference, in which we use data to draw conclusions about a general situation or population of which the data can be considered randomly representative. Example E: Where does your newspaper land. Example F: Which seedling will appear first. Probabilities are fractions of the whole probability of 1. What would we need to assume for these 5 areas to be equally likely places for the newspaper to land. If the box is well shaken, then none of these colours is more likely than the other. Notice that a very very very small chance of something happening is not a zero chance. Can we assume they are equally likely. They have seen variation in results of simple chance experiments.

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Events used for some games of chance. Situations involving uncertainty or randomness include probability in their models, and analysis of models often leads to data investigations to estimate parts of the model, to check the suitability of the model, to adjust or change the model, and to use the model for predictions. The probability of a blue one is zero. The concepts and tools of probability pervade analysis of data. Can probability be 0. Example C: What colour sweet did you get. Example F: Which seedling will appear first. Example A: Throwing one die. Someone tells us that the possible colours for the stars of a Roman candle are red, blue, green, and yellow and we spread our total probability of one over those four colours in guessing which colour is going to shoot highest. Comparisons of probabilities − which are equal, which are not, how much bigger or smaller − are therefore also of interest in modelling chance. For example, the list of possible colours might be red, yellow, green, blue, orange, brown. Otherwise we haven’t considered all possibilities. So probabilities for the possibilities of a situation are parts of the whole probability of 1. If there’s only 1 blue sweet and lots of others, there’s not much chance the blue will be shaken out, but there’s still some chance. In Year 5, consideration of the possible outcomes and of the circumstances of simple situations has lead to careful description of the outcomes and assumptions that permit assigning equal probabilities, understanding what the values of these probabilities must be and representing these values using fractions. In the spinner pictured below, there are 4 colours. First we need to decide what to do if the arrow appears to land on one of the dividing lines. Provided there are some blue sweets in the box, there is some chance a blue sweet will be shaken out. We didn’t make a mistake; we just didn’t have full information. The concepts and tools of probability pervade analysis of data.

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Statistics and statistical thinking have become increasingly important in a society that relies more and more on information and calls for evidence. Example D: Which colour firework goes the highest. Example D: Which colour firework goes the highest. In Year 5, consideration of the possible outcomes and of the circumstances of simple situations has lead to careful description of the outcomes and assumptions that permit assigning equal probabilities, understanding what the values of these probabilities must be and representing these values using fractions. Probability is a measure, like length or area or weight or height, but a measure of the likeliness or chance of possibilities in some situation. 1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 4, 4, 1, 1, 5, 5, 1, 1, 6, 6, 1, 2, 2, 2, 3, 3, 2, 2, 4, 4, 2, 2, 5, 5, 2, 2, 6, 6, 2, 3, 3, 3, 4, 4, 3, 3, 5, 5, 3, 3, 6, 6, 3, 4, 4, 4, 5, 5, 4, 4, 6, 6, 4, 5, 5, 5, 6, 6, 5, 6, 6. Example D: Which colour firework goes the highest. Example D: Which colour firework goes the highest. A list of the possible landing places for the newspaper is footpath but not the driveway, driveway on the footpath, driveway inside the boundary, grass inside the boundary, garden inside the boundary. Otherwise we haven’t considered all possibilities. In the spinner pictured below, there are 4 colours. This work is licensed under the Creative Commons Attribution NonCommercial NoDerivs 3. Example D: Which colour firework goes the highest. From Years 1 4, students have gradually developed understanding and familiarity with simple and familiar events involving chance, including possible outcomes and whether they are “likely”, “unlikely” with some being “certain” or “impossible”. Any interpretation of data involves considerations of variation and therefore at least some concepts of probability. If the arrow spins smoothly and if the spin is well done so that the arrow spins sufficiently thoroughly that it “forgets” its starting point, then each of the colours is equally likely to be the outcome. Concepts of probability underpin all of statistics, from handling and exploring data to the most complex and sophisticated models of processes that involve randomness. Any interpretation of data involves considerations of variation and therefore at least some concepts of probability. Each of these therefore is. Usually this is done by someone driving around the streets and throwing rolled and covered newspapers into the front of people’s places. Can we assume they are equally likely. Hence the need to develop statistical skills and thinking across all levels of education has grown and is of core importance in a century which will place even greater demands on society for statistical capabilities throughout industry, government and education. We will consider two different coloured dice or two dice that are marked in some way so that we can tell them apart. Suppose you have a small box of different coloured sweets, such as MandM’s or Smarties. But if there are no blue sweets at all in the box, then there’s no chance of a blue − it’s impossible to get a blue if there are no blues in the box. Sometimes it is very easy to describe possible events and sometimes there is really only one way of describing them, but in many situations this is not so and careful description is therefore important. In each of the above examples, we have been able to describe the possible outcomes of a situation in terms of a complete list of outcomes, and been able to identify circumstances under which it is reasonable to assume that none of these outcomes are more or less likely than the others, that is, that all of these outcomes are equally likely. Example E: Where does your newspaper land. In Year 5, consideration of the possible outcomes and of the circumstances of simple situations has lead to careful description of the outcomes and assumptions that permit assigning equal probabilities, understanding what the values of these probabilities must be and representing these values using fractions.

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If one or more faces are more likely to come up than others, the die is called a “loaded” die. This work is licensed under the Creative Commons Attribution NonCommercial NoDerivs 3. There is nothing wrong with saying that this is what we are assuming, as long as we do say it clearly. Any interpretation of data involves considerations of variation and therefore at least some concepts of probability. Example C: What colour sweet did you get. But is this sensible. Of course there must be at least one of each of the colours in the firework, but do we need to have equal number of the colours in the firework. In this module, in the context of understanding chance in everyday life, we build on the preliminary concepts of chance of Years 1 4 to focus more closely on describing possible events in simple and everyday situations in which it is reasonable to assign equal probabilities to outcomes. Notice that a very very very small chance of something happening is not a zero chance. The objectives of the chance and probability strand of the F 10 curriculum are to provide a practical framework for experiential learning in foundational concepts of probability for life, for exploring and interpreting data, and for underpinning later developments in statistical thinking and methods, including models for probability and data. First we need to decide what to do if the arrow appears to land on one of the dividing lines.

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Therefore each bit of probability is. In Year 6, decimals and percentages are used along with fractions to describe probabilities. In the spinner pictured below, there are 4 colours. Again it is simple to describe the basic outcomes of throws of a pair of dice, whether they are tossed together or one after the other. We have planted 10 seeds at regular places perhaps using a tray with 10 separate small compartments in a planter box at the same time, and carefully watered them each day. Probability is a measure, like length or area or weight or height, but a measure of the likeliness or chance of possibilities in some situation. You give one to your friend by shaking one out of the box onto your friend’s hand. And then during the firework display, a purple one shoots out. Example F: Which seedling will appear first. However, it is more difficult in practice to toss a coin completely randomly than to toss dice randomly, and also to assume that a particular person does not tend to toss a coin in a similar manner each toss. Simple everyday events. Each time we observe the situation, only one possible outcome can be observed. We start with very simple situations that are part of games of chance, and then consider some simple everyday events. This situation is far more difficult than is usually assumed in such research which often tends to take the assumptions of fair coins and independent tosses as absolutely immutable and not to be questioned. The probability of a blue one is zero. The list of possible outcomes is the list of pairs of numbers from 1 to 6, where the first number in each pair is the uppermost face of die 1 for example, this might be a red die and the second number in each pair is the uppermost face of die 2 for example, this might be a blue die as follows. For example, if a coin is fair and tosses are independent, the chance of HTHHTT is EXACTLY the same as the chance of HHHHHH, but many people think of the first outcome in terms of getting 3 H’s and 3 T’s which is not the same as getting a particular sequence with 3 H’s and 3 T’s. How can we “find” values.

He has made a number of television appearances, on such shows as “Good Morning America” and “48 Hours ” He’s also been interviewed on many radio shows in the New York City area, including the Dick Summers show, the Lee Speigle show, Lloyd Strayhorn’s “Numbers and You,” and the Candy Jones Show

In the list of the possible landing places for the newspaper of footpath but not the driveway, driveway on the footpath, driveway inside the boundary, grass inside the boundary, garden inside the boundary, have we considered all possibilities. In throwing one die, there are 6 possible outcomes. What is the chance you will shake out a blue sweet from your box onto your friend’s hand. Example C: What colour sweet did you get. Under these carefully described circumstances and assumptions, the probability of each of the list of outcomes is 1/total number of outcomes in the list. Probability models are at the heart of statistical inference, in which we use data to draw conclusions about a general situation or population of which the data can be considered randomly representative. What do we need to know or assume, to be able to say that the highest star is equally likely to be any of those 5 colours. They have seen variation in results of simple chance experiments. They have seen variation in results of simple chance experiments. This situation is far more difficult than is usually assumed in such research which often tends to take the assumptions of fair coins and independent tosses as absolutely immutable and not to be questioned. In Year 4, they have considered which events are more or less likely, and, if events are not more or less likely than others, then they have considered that it is reasonable to assume the events to be equally likely. There may be a very very very small chance that it will snow in Brisbane but it’s not absolutely 0. The concepts are experienced through examples of situations familiar and accessible to Year 5 students and that build on concepts introduced in 1 4. Someone tells us that the possible colours for the stars of a Roman candle are red, blue, green, and yellow and we spread our total probability of one over those four colours in guessing which colour is going to shoot highest. The basic events are pairs of numbers, where each number is the face value of the uppermost face of one of the dice. For example, the list of possible colours might be red, yellow, green, blue, orange, brown. A fair die is one for which it is assumed that the 6 faces are equally likely to be uppermost in a toss of the die.