Probability

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. Alternatively, we could say that we are only considering the situation of the newspaper landing on the footpath or front area of our property, so that we are restricting the situation to that to exclude the possibility of it landing in the road, gutter or missing our place. 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. 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. 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. Example C: What colour sweet will you get. 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 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. In Year 6, decimals and percentages are used along with fractions to describe probabilities. The possible events here are the possible colours of the sweet that lands in your friend’s hand. 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. Statistics is the science of variation and uncertainty. In addition a common mistake is to state that it demonstrates either poor or, in contradiction, good thinking, to say that a particular sequence of coin tosses with a mixture of heads and tails is more likely than a sequence of the same outcome, when in practice it is not made clear whether the researcher or the student is thinking of a particular ordered sequence or merely a mixture of heads and tails. How will real observations compare with what we expect to see. Although it is space xy crash not completely clear how this affects the chances of the highest colour, the assumption of equal numbers of stars in each colour does tend to make it more believable that the highest star is equally likely to be any colour because then we are just saying that the heights of the stars are random and are not affected by any of the other stars. 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. 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. Some fireworks used in public fireworks displays send up coloured stars or flashes. 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. 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. 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. The possible colours of the sweet you shake from the box into your friend’s hand are these 5 colours. To see an example of contradictions in research approaches that take the assumptions of fair coin and independent tosses as absolute and not to be questioned, contrasted with use of observed sequences as learning experiences for statistical hypothesis testing, compare Ridgway and Ridgway 2010, Teaching Statistics Vol 32, number 2, p34, with Holland 2007, Teaching Statistics Vol 29, number 3, p71. Many board games use throws of two dice. How much is all the probability. They have seen variation in results of simple chance experiments.

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We didn’t make a mistake; we just didn’t have full information. 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. 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. Then for a finite number of equally likely outcomes of a simple and everyday situation, we can assign the equal probabilities as fractions. The concepts are experienced through examples of situations familiar and accessible to Year 5 students and that build on concepts introduced in 1 4. Before thinking about the chances of events, we need to identify and describe them clearly so that there is no confusion and anyone reading or hearing our description will be thinking of the events in the same way as us. Assumed background from 1 4. Some general comments and links from F 4 and towards year 6. Otherwise we haven’t fully described the situation. How can we “find” values. Statistics and Probability : Module 10Year : 5. Simple everyday events. 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. 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. They have seen variation in results of simple chance experiments. We can estimate them from data. So the list of the possible outcomes of a single spin of the arrow is orange, red, green, blue. 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. Example D: Which colour firework goes the highest. 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. In addition a common mistake is to state that it demonstrates either poor or, in contradiction, good thinking, to say that a particular sequence of coin tosses with a mixture of heads and tails is more likely than a sequence of the same outcome, when in practice it is not made clear whether the researcher or the student is thinking of a particular ordered sequence or merely a mixture of heads and tails. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. What is the chance you will shake out a blue sweet from your box onto your friend’s hand. Example E: Where does your newspaper land. Another possibility is to say that if the arrow appears to fall on a line, take it as falling on the colour to the left of the line. Otherwise we haven’t considered all possibilities.

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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. There may be a very very very small chance that it will snow in Brisbane but it’s not absolutely 0. 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. Therefore each bit of probability is. 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. 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. 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. Usually in these types of very simple situations, we can also easily list these simple events. Another possibility is to say that if the arrow appears to fall on a line, take it as falling on the colour to the left of the line. First we need to decide what to do if the arrow appears to land on one of the dividing lines. Therefore each bit of probability is. Statistics and statistical thinking have become increasingly important in a society that relies more and more on information and calls for evidence. Assumed background from 1 4. Note this only applies not only when we can consider events as equally likely but also when we can give a complete list the outcomes, of which there are a fixed and known total number. Another possibility is to say that if the arrow appears to fall on a line, take it as falling on the colour to the left of the line. So we need to divide our total probability of 1 into 6 equal bits of probability. 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”. 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. The concepts and tools of probability pervade analysis of data. If we assume this, then the total probability of 1 is divided into 10 equal bits of probability, so that each place has a probability of to have the first shoot of green appearing. 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.

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Example E: Where does your newspaper land. So probabilities for the possibilities of a situation are parts of the whole probability of 1. So the total probability of 1 is divided into 4 equal bits of probability. 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. 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. Many board games use throws of two dice. 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. We can estimate them from data. Commentators often like to ask crowds watching to guess which colour star will be next or which colour star will go the highest. Provided there are some blue sweets in the box, there is some chance a blue sweet will be shaken out. For some excellent comments on this, see the letter by Harvey Goldstein to the Editor of Teaching Statistics 2010, Volume 32, number 3.

Legacy

Below are some everyday situations for which it is easy to list the possible outcomes. 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. The possible events here are the possible colours of the sweet that lands in your friend’s hand. 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. The total probability of 1 is divided into 36 equal bits of probability, each being. Example E: Where does your newspaper land. 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. If the areas of those 5 places are equal and if the delivery person throws at random over those 5 places then we can assume they are equally likely. Much is written in educational literature about coin tosses and much has been tried to be researched and analysed in students’ thinking about coin tossing. 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. Concepts of probability underpin all of statistics, from handling and exploring data to the most complex and sophisticated models of processes that involve randomness. Note this only applies not only when we can consider events as equally likely but also when we can give a complete list the outcomes, of which there are a fixed and known total number. First we need to decide what to do if the arrow appears to land on one of the dividing lines. 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.

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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. An extension to events used for some games of chance. Some get a bigger parcel of probability and some get a smaller parcel. 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. Some get a bigger parcel of probability and some get a smaller parcel. Usually in these types of very simple situations, we can also easily list these simple events. This work is licensed under the Creative Commons Attribution NonCommercial NoDerivs 3. How will real observations compare with what we expect to see. 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. And then during the firework display, a purple one shoots out. 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.

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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. The basic events are pairs of numbers, where each number is the face value of the uppermost face of one of the dice. Although it is not completely clear how this affects the chances of the highest colour, the assumption of equal numbers of stars in each colour does tend to make it more believable that the highest star is equally likely to be any colour because then we are just saying that the heights of the stars are random and are not affected by any of the other stars. We will consider situations in which the description of the events is straightforward and natural and easily listed. In throwing one die, there are 6 possible outcomes. We will consider two different coloured dice or two dice that are marked in some way so that we can tell them apart. 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. What would we need to assume for these 5 areas to be equally likely places for the newspaper to land. There are 36 of these possible pairs. How will real observations compare with what we expect to see. The possible events here are the possible colours of the sweet that lands in your friend’s hand. When spun around its pivot, the arrow of the spinner comes to rest on a segment of a circle. Probability is a relative measure; it is a measure of chance relative to all the other possibilities of the situation. We are saying that the thrower won’t miss those 5 places, but the throw is random within that restriction. So we take all the probability and parcel it out amongst the various possibilities. Example D: Which colour firework goes the highest. Concepts of probability underpin all of statistics, from handling and exploring data to the most complex and sophisticated models of processes that involve randomness. Appendix Note for teachers’ background information. Example F: Which seedling will appear first. The Improving Mathematics Education in Schools TIMES Project 2009 2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. Each of these therefore is. Events used for some games of chance. What is the chance you will shake out a blue sweet from your box onto your friend’s hand. The concepts and tools of probability pervade analysis of data.

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Example D: Which colour firework goes the highest. 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. Note this only applies not only when we can consider events as equally likely but also when we can give a complete list the outcomes, of which there are a fixed and known total number. Any interpretation of data involves considerations of variation and therefore at least some concepts of probability. Example E: Where does your newspaper land. 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. Some fireworks used in public fireworks displays send up coloured stars or flashes. Statistics and Probability : Module 10Year : 5. The total probability of 1 is divided into 36 equal bits of probability, each being. Probabilities are fractions of the whole probability of 1. Many board games use throws of one or two dice to give how many places to move or what choice is to be made. Probability is a relative measure; it is a measure of chance relative to all the other possibilities of the situation. Simple everyday events. In some games that use a spinner, if the arrow appears to fall on a line, the spin is taken again. Probability is a relative measure; it is a measure of chance relative to all the other possibilities of the situation. 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. This work is licensed under the Creative Commons Attribution NonCommercial NoDerivs 3. Therefore, it is very important to be clear about the situation being considered. However, even though considerations of probability pervade all of statistics, understanding the results of some areas of data analysis requires only basic concepts of probability. Usually the different segments are in different colours, so the basic events of a single spin are the colours used for the different segments. 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. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. 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. So the total probability of 1 is divided into 4 equal bits of probability. 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 relative measure; it is a measure of chance relative to all the other possibilities of the situation. The Improving Mathematics Education in Schools TIMES Project 2009 2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. How can we “find” values. Otherwise we haven’t fully described the situation. In many games, such as board games, moves are decided by the chance throws of dice or spins of spinners. So under these conditions, each of the 36 possible outcomes is equally likely. So we take all the probability and parcel it out amongst the various possibilities. 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. Many board games use throws of one or two dice to give how many places to move or what choice is to be made. Statistics and Probability : Module 10Year : 5. If the box is well shaken, then none of these colours is more likely than the other. They have looked at simple everyday events where one cannot happen if the other happens, and, in contrast, they have thought about simple situations where it is reasonable to assume that the chance of occurrence of an event is not changed by the occurrence of another. 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.