Chance and Data

1. Chance and Data summary

2. Probability Fundamentals 1. A probability is calculated by; no# of interest / no# of possible E.g. there are 10 marbles in a bag. 2 are blue, 5 are red and 3 are yellow. What is the probability of selecting a blue marble; No# of interest = 2 blue marbles No# of possible = 10 marbles So Probability = This could also be written as; (a simplified fraction) or 0.2 (a decimal) or 20% (a percentage) When you are asked about 2 different probabilities you will be asked about either: 2. Prob.1 AND Prob.2 – here you multiply the probabilities E.g. What’s the probability of picking a redANDa yellow marble from the bag above? Ans; prob(red) = and prob(yellow) = so 3. Prob.2 OR Prob.2 – here you add the probabilities E.g. What is the probability of selecting a redOR a yellow marble from the bag above? Ans;

3. Multivariate Data Tables • These are when data with more than one variable is shown on one table. E.g. • This shows the probabilities of different combinations occurring. E.g. there would be chance of randomly selecting a red magazine from all the books. Or a chance of randomly selecting a textbook from all of the red books.

4. Probability Trees This is when there is one variable, but more than one event occurring. • The 1st set of branches are the options of the 1st first decision and the 2nd set of branches are the options of the 2nd decision. • We multiply the probability of the 1st option by the 2nd option to give the overall probability. E.g. There are 10 marbles in a bag; 5 are red, 3 are blue and 2 are yellow. Draw a probability tree to show the probability of pulling out two marbles of the same colour: The second event is (again) to pick out a red, blue or yellow marble (there are only 9 marbles left now) The first event is to pick out a red, blue or yellow marble

5. Analysing Statistical Data You will need to write comments/analysis about: • Statistical enquiries (someone has taken a sample, done an investigation and drawn conclusions). • Data displayed using box and whisker graphs • Data showing trends from other graphs • Other chance related situations (such as a raffle or an advertisement claiming ‘1 in 3 is a winner!’) Each time you should follow the same pattern in your explanation: • Define or list the problems/issues around what you have read • Explain how you know this using your understanding of Stats (this could be mean, median, mode, probability calculations, range, upper/lower quartile, inter-quartile range etc.) • Discuss how you could improve the situation, which assumptions need to change, or in which situations would they be correct.

6. 1. Statistical enquiries • You can comment on the following THREE sections: • Their purpose and planning You want to comment on whether you think their sample was an accurate representation of the population. Did the selection process mean that their sample would give good results? • Their analysis Comment on the graphs or charts that were used to show the data and the important features(for example, a box and whisker will show more relevant data than a scatter graph). • Their conclusions Comment on whether the conclusion was fair, accurate and whether was it justified. You will need to explain that there could be better conclusions (e.g. a median is better measure of average than the mode). You will then need to explain (justify) why you think this (e.g. mode does not necessarily give a good indication of the ‘middle.’)

7. 2. Data Displayed using Box and Whisker Graphs • Comment on the Relevant features. • Generally speaking: • The most relevant measure of average is MEDIAN (middle, is not affected by outliers) • The most relevant measure of spread is INTER-QUARTILE RANGE (not affected by outliers, shows how spread out the results are) • Min, Max and Range are (generally) NOT good measures and do not give accurate results

8. 3. Data Showing Trends of Graphs • There will be a trend of the graph which may or may not be accurate enough to draw conclusions. You need to comment on whether there is a strong, medium or weak correlation between the data and the line of best fit Identify features that would affect the accuracy of the data, such as clusters or outliers. Cluster Medium/weak correlation

9. 4. Chance Related Situations • This will be some random situation that involves a probability • Do a calculation to show what the actual probability will be.In this case; So there is a one in 216 chance of winning (not one in 50) • Explain why you did the calculation. • Discuss how the situation could be corrected (change the ad) and the underlying assumptions (is the dice 6 sided? Is it weighted? Etc.) $5 to enter the draw. Simply roll three 6’s in a row. You have a one-in-fifty chance of winning a BRAND NEW IPAD!!

10. Remember: • In your writing – keep a structure! Follow the steps; (1) Define the problem. (2) Explain the definition and show calculations. (3) Discuss and justify potential improvements • Merit and Excellence questions can be a combination of these different ideas (grouped into a single question). Learn these slides clearly so that you can spot what the examiner is asking you in the exam.