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Making Social Work Count Lecture 8 An ESRC Curriculum Innovation and Researcher Development Initiative

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Probability and Chance How to make a decision with confidence

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What are the chances that someone in alcohol treatment will experience a slip or relapse in their alcohol use if they occasionally used illegal drugs prior to treatment? What are the chances something will happen?

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Learning outcomes

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From storytelling to decision-making 1.observing 2.categorising and charting 3.summarising and storytelling 1.theorising and hypothesizing 2.testing 3.deciding and predicting

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What are the chances?

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What are the chances that you will use research to influence your social work practice? What’s the probability that a service user will remain abstinent from alcohol after treatment? How likely is an adult service user going to receive local authority services if assessed as ‘low’ need? How likely is it that you will pass this class?

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Most likely Probably Pretty good chance No chance Not at all Definitely Unlikely

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Probability How likely something is to happen What is the number of times that a specific event can occur? What is the number of times that any event can occur?

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Probability Probability (P) varies from 0 to 1.0 Can be represented as a decimal (1.0) or a percentage (100%). 0.1.2.3.4.5.6.7.8.9 1.0 ImpossibleCertainty

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Probability P =.50 or 50% chance 0.1.2.3.4.5.6.7.8.9 1.0 ImpossibleCertainty

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Probability # of times the outcome or event can occur total # of times any outcome or event can occur (sample space) P =

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The coin toss What is the probability of getting heads on a coin toss?

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The coin toss 2 possible outcomes (sample space) Heads Tails Possibility of getting a heads on a toss? ½ =.50 (50%)

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Jelly beans What is the sample space? What is the P of selecting a: – red jelly bean out of a bag? – black jelly bean out of a bag? – green jelly bean out of a bag? – blue jelly bean out of a bag?

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Jelly beans sample space = 9 Different values of P – P (red) = 1/9 =.11 (11%) – P (black) = 2/9 =.22 (22%) – P (green) = 3/9 =.33 (33%) – P (blue) = 0/9 = 0 (0%)

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What are the rules?

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Converse rule the probability of an event NOT occurring Event NOT occurring = 1 – P What is the P of NOT selecting a red jelly bean out of the bag? What is the P of NOT selecting a blue jelly bean out of the bag?

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Converse rule the probability of an event NOT occurring Event NOT occurring = 1 – P Not selecting a red jelly bean = 1 -.11 =.89 (89%) Not selecting a blue jelly bean = 1 – 0 = 1.00 (100%)

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Addition rule The probability of obtaining more than one outcome (outcome A OR outcome B) equals the sum of their separate probabilities. The two outcomes need to be mutually exclusive. P of selecting A or B = P (event A) + P (event B) What is the P of selecting a red or black jelly bean out of the bag? What is the P of selecting a red, black or green jelly bean out of the bag?

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Addition rule The probability of obtaining more than one outcome (outcome A OR outcome B) equals the sum of their separate probabilities. The two outcomes need to be mutually exclusive. P of selecting A or B = P (event A) + P (event B) P of selecting red or black =.11 +.22 =.33 (33%) P of selecting red, black or green =.11 +.22 +.33 =.66 (66%)

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Multiplication rule the probability of obtaining two or more outcomes at the same time equals the product of their separate probabilities P of obtaining A and B outcomes = P (event A) x P (event B) What is the P of obtaining heads on both of two coin flips?

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Multiplication rule the probability of obtaining two or more outcomes at the same time equals the product of their separate probabilities P of obtaining A and B outcomes = P (event A) x P (event B) P of obtaining heads on both flips =.50 x.50 =.25 (25%)

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Probability of a particular level of need Adult individuals assessed as: Probability represented as a fraction Probability represented as a decimal and % Low: 100 Moderate: 250 Critical: 500 Substantial: 150 Total: 1000 1.What is the probability of an individual NOT being assessed as moderate? 2.What is the probability of an individual being assessed as either critical OR substantial?

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Probability of a particular level of need Adult individuals assessed as: Probability represented as a fraction Probability represented as a decimal and % Low: 100100/10000.1 (10%) Moderate: 250250/10000.25 (25%) Critical: 500500/10000.5 (50%) Substantial: 150150/10000.15 (15%) Total: 1000 1.What is the probability of an individual NOT being assessed as moderate? 1 – 0.25 = 0.75 (75%) 2.What is the probability of an individual being assessed as either critical OR substantial? 0.5 + 0.15 =.20 (20%)

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? How is probability related to theory or hypothesis?

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Probability distributions Frequency distributions (observed data) Reason f (%) Physical disability 160 (51%) Temp. ill 144 (46%) Mental health 7 (.3%) Total N = 311 (100%) Sex f (%) Male 112 (36%) Female 199 (64%) Total N = 311 (100%) Probability distributions (theoretical outcomes) Outcome P (%) Red.11 (11%) Orange.11 (11%) White.11 (11%) Yellow.11 (11%) Black.22 (22%) Green.33 (33%) Total = 1.00 (100%)

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Probability distribution Construct a probability distribution for obtaining heads on one flip of a coin HeadsP (%) 0 1 Total = Construct a probability distribution for obtaining heads on two flips of a coin HeadsP (%) 0 1 2 Total =

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Probability distribution Construct a probability distribution for obtaining heads on one flip of a coin HeadsP (%) 0.50 (50%) 1.50 (50%) Total = 1.00 (100%) Construct a probability distribution for obtaining heads on two flips of a coin HeadsP (%) 0.25 (25%) 1.50 (50%) 2.25 (25%) Total = 1.00 (100%)

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Exercise: Does the theory hold up? Flip a coin twice and record the number of heads Repeat the coin toss for a total of ten times and record the number of heads Develop a frequency distribution based on your observed data Headsf (%) 0 1 2 Total =

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Do more observations lead to perfection? 10 flips Headsf (%) 04 (40%) 14 (40%) 22 (20%) Total = 10 (100%) 1000 flips Headsf (%) 0253 (25.3%) 1499 (49.9%) 2248 (24.8%) Total = 1000 (100%)* *Taken from Levin, J., & Fox, J.A. (2003). Elementary statistics in social research (9 th ed.). Boston: Allyn and Bacon.

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Do more observations lead to perfection? 10 flips Headsf (%) 04 (40%) 14 (40%) 22 (20%) Total = 10 (100%) 1000 flips Headsf (%) 0253 (25.3%) 1499 (49.9%) 2248 (24.8%) Total = 1000 (100%) Heads P (%) 0.25 (25%) 1.50 (50%) 2.25 (25%) Total = 1.00 (100%)

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How accurate was our probability of assessing a particular level of need? Level of need2008 – 2012P (%) Low300.06 (6%) Moderate2125.425 (42.5%) Critical2025.405 (40.5%) Substantial550.11 (11%) Total5000 Adult individuals assessed as: Probability represented as a fraction Probability represented as a decimal and % Low: 100100/10000.1 (10%) Moderate: 250250/10000.25 (25%) Critical: 500500/10000.5 (50%) Substantial: 150150/10000.15 (15%) Total: 1000

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? Is something truly representative (accurate) or has it occurred by chance?

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The theory of the Normal Curve Probability distribution or normal distribution represented graphically How the data would most likely be distributed in a real world Is symmetrical and unimodal Mean, median and mode coincide at the middle of the curve Used to describe distributions and make statements of P

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Characteristics of the Normal Curve

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Use of the Normal curve To understand general patterns in a population – What is normal (average) and are there any extreme cases? Most sample data is imperfect, that is, it may not accurately reflect the true population. – The sample may be skewed by several extreme cases The normal curve is the theoretical curve of all possible samples for a variable – it is the mean of all possible sample means, which equals the mean of the population.

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Not all of the world is a Normal curve mean, median and mode mode median mean

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The Normal curve as theoretical ideal To what extent does the data collected by researchers (through sampling the population) reflect or approximate a normal curve or the true mean of the population? As the number of observations increase, the more likely the data will represent a normal curve and the true population mean The normal curve assumes that all social, psychological and physical data are normally distributed Is used to help predict how likely the sample data accurately reflects the true population mean

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How accurate was our coin toss exercise? Probability 10 flips 1000 flips

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The area under the normal curve 68.26% of data 95.44% of data 99.74% of data 34.13% 47.72% 49.87%

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Social Work Salaries £19,246 £21559 £23872 £26,185 £28498 £30811 £33,124

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Social Work Salaries Looking at the distribution of salaries, answer the following questions: – What is the mean salary of social workers? – How likely is it that you will make between £23,872 and 28,498? – How likely is it that you will make over £33,124? – What could be contributing factors to the differences in the salaries?

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Social Work Salaries £19,246 £21,559 £23,872 £26,185 £28,498 £30,811 £33,124 68.26% 95.44% 99.74%

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Learning outcomes Are you able to: Define and discuss the theory and rules of probability Calculate probability and create a probability distribution with example data Describe the characteristics of a normal curve and interpret a normal curve using example data

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Activity

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Activity Ask the students to read the following scenario: You are a social worker working in a community mental health team. You are working with Adele (36-years-old, White British) who has been diagnosed with a Personality Disorder and alcohol dependency. Adele is married, employed and has two children. Her recent alcohol misuse has greatly interfered with her employment and social life and her husband has threatened to leave her and take the two children with him if she does not address her alcohol use. Adele has also disclosed to you that she occasionally uses cocaine. Adele has volunteered to enter a “total abstinence” 28-day treatment programme and you have supported her in entering the programme. You have also recently read the following research article: Charney, D.A., Zikos, E., & Gill, K.J. (2010). Early recovery from alcohol dependence: factors that promote or impede abstinence. Journal of Substance Abuse Treatment, 38, 42-50.

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Activity – continued The article addresses the probability and likelihood of remaining abstinent following a 4-week treatment programme and the factors that might impede abstinence. Based on Adele’s personal characteristics and social situation and the findings of this study, answer the following questions: What would you say is the likelihood that Adele may “slip” or “relapse” during the first 4 weeks of treatment? What information and evidence from the article did you use to answer this question? What would you say is the likelihood that Adele will remain abstinent after treatment if she experiences a “slip” or “relapse” during treatment? As her social worker, based on the findings from this research, what resources and interventions could you put into place to increase her chances of remaining abstinent during and after treatment? How could you help to prepare Adele for the future? Discuss the limitation of this research.

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References Charney, D.A., Zikos, E., & Gill, K.J. (2010). Early recovery from alcohol dependence: factors that promote or impede abstinence. Journal of Substance Abuse Treatment, 38, 42-50. Herrenkohl, T.I., Sousa, C., Tajima, E.A., Herrenkohl, R.C., & Moylan, C.A. (2008). Intersection of child abuse and children’s exposure to domestic violence. Trauma, Violence & Abuse, 9, 84-99. – Lecturers can refer to this article to discuss the impact of exposure to domestic violence and child abuse on children and the likelihood of various outcomes. Holt, S., Buckley, H., & Whelan, S. (2008). The impact of exposure to domestic violence on children and young people: a review of the literature. Child Abuse & Neglect, 32, 797-810. – Lecturers can use this literature review to provide a review of the evidence of the impact of exposure to domestic violence on children and the likelihood of various outcomes. Galvani, S., Dance, C., & Hutchinson, A. (2013). Substance use training experiences and needs: findings from a national survey of social care professionals in England. Social Work Education, 32, 888-905. – Lecturers could refer to this article when discussing distribution and the normal curve (page 899).

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