# Leaving Certificate - Probability and Statistics Higher Level

## Description

Probability and Statistics is one of two strands introduced in the first phase of the new Project Maths Course in the Irish curriculum. This topic covers up to half of the new Paper 2 in the Leaving Certificate Paper.

Statistics are used in real life to make sense of the information around us and how it affects us. Statistics looks at the data handling cycle and analysis of the data collected. This involves posing a question, collecting data on that question, presenting that data, analysing the data (using measures of spread and centre) and interpreting the results. In answering questions, it is essential that you can contextualise and justify your findings.

Probability is concerned with the likelihood of an event(s) happening. The information can be used to make informed decisions. The use of probability is commonly utilised in the world of finance, insurance and sport among others. Probability can also be used to infer the fairness of an event or series of events. It can be evaluated using a diagram or a rule-based approach. A combination of Probability and Statistics can be used to prove/disprove a given conjecture or statement (Hypothesis Testing (HL only)).

This Strand attempts to merge the mathematical aspects of Probability and Statistics with its real-life application. It is an interesting topic that is very accessible to all students.

## Learning Outcomes

**Students will learn about:****1.1 Counting**

- - count the arrangements of n distinct objects (n!)
- - count the number of ways of arranging r objects from n distinct objects
- - count the number of ways of selecting r objects from n distinct objects

**1.2 Concepts of probability**

- - discuss basic rules of probability (AND/OR, mutually exclusive) through the use of Venn diagrams
- - calculate expected value and understand that this does not need to be one of the outcomes
- - recognise the role of expected value in decision making and explore the issue of fair games
- - extend their understanding of the basic rules of probability (AND/OR, mutually exclusive) through the use of formulae
- - use the Addition Rule, Multiplication Rule (Independent events), Multiplication Rule (General case)
- - solve problems involving conditional probability in a systematic way

**1.3 Outcomes of random processes**

- - find the probability that two independent events both occur
- - apply an understanding of Bernoulli trials
- - solve problems involving up to 3 Bernoulli trials
- - calculate the probability that the 1st success occurs on the nth Bernoulli trial where n is specified
- - solve problems involving calculating the probability of k successes in n repeated Bernoulli trials (normal approximation not required)
- - calculate the probability that the kth success occurs on the nth Bernoulli trial
- - use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions
- - solve problems involving reading probabilities from the normal distribution tables

**1.4 Statistical reasoning with an aim to becoming a statistically aware consumer**

- - work with different types of bivariate data

**1.5 Finding, collecting and organising data**

- - discuss different types of studies: sample surveys, observational studies and designed experiments
- - design a plan and collect data on the basis of above knowledge
- - recognise the importance of randomisation and the role of the control group in studies
- - recognise biases, limitations and ethical issues of each type of study
- - select a sample (stratified, cluster, quota – no formulae required, just definitions of these)
- - design a plan and collect data on the basis of above knowledge

**1.6 Representing data graphically and numerically****1.6a Graphical**

- - describe the sample (both univariate and bivariate data) by selecting appropriate graphical or numerical methods
- - explore the distribution of data, including concepts of symmetry and skewness
- - compare data sets using appropriate displays, including back-to-back stem and leaf plots
- - determine the relationship between variables using scatterplots
- - recognise that correlation is a value from -1 to +1 and that it measures the extent of the linear relationship between two variables
- - match correlation coefficient values to appropriate scatter plots
- - understand that correlation does not imply causality
- - analyse plots of the data to explain differences in measures of centre and spread
- - draw the line of best fit by eye
- - make predictions based on the line of best fit
- - calculate the correlation coefficient by calculator

**1.6b Numerical**

- - recognise standard deviation and interquartile range as measures of variability
- - use a calculator to calculate standard deviation
- - find quartiles and the inter-quartile range
- - use the interquartile range appropriately when analysing data
- - recognise the existence of outliers
- - recognise the effect of outliers
- - use percentiles to assign relative standing

**1.7 Analysing, interpreting and drawing inferences from data**

- - interpret a histogram in terms of distribution of data
- - make decisions based on the empirical rule
- - recognise the concept of a hypothesis test
- - calculate the margin of error for a population proportion
- - conduct a hypothesis test on a population proportion using the margin of error

**1.8 Synthesis and problem-solving skills**

- - explore patterns and formulate conjectures
- - explain findings
- - justify conclusions
- - communicate mathematics verbally and in written form
- - apply their knowledge and skills to solve problems in familiar and unfamiliar contexts
- - analyse information presented verbally and translate it into mathematical form
- - devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions

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