MEI Conference Session Title. Presenter Terry Dawson. An introduction to probability distributions

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1 MEI Conference 2016 Session Title An introduction to probability distributions Presenter Terry Dawson Presenter address

3 Tree diagram representing rolling 4 dice Key p = P(Get a 5) q = P(Don t get a 5) Outcome Notation Probability 4 fives P 4 3 fives P 3 q 2 fives 1 five 0 fives

4 Making connections 1. How many times did each of the outcomes below in the tree diagram? p 4 p 3 q p 2 q 2 pq 3 q 4 You may have seen expressions like the outcomes above before, when expanding expressions like: (x+y) 4 = x 4 +4x 3 y+6x 2 y+4x 3 y+y 4 Think back to how you were able work out the coefficients of the terms in the expansion of expressions like (x+y) n. 2. Look at the coefficients of the terms in the expansion of (x+y) 4 and the number of occurrences p 4, p 3 q, p 2 q 2, pq 3 and q 4. What do you notice? 3. If we let p = (1/6) the probability of getting a 5 and q = (5/6) the probability of not getting a 5 on each trial (roll of the dice). Work out the value of: p 4 4p 3 q 6p 2 q 2 4pq 3 q 4 4. Compare these values with the theoretical probabilities in your table. 5. Using the theoretical probabilities answer the questions below. (a) On average, how many times would you expect to get 4 fives if you roll 4 dice 40 times? (b) On average, how many times would you expect to get 3 or more fives if you roll 4 dice 40 times? (c) When rolling 4 dice 40 times, on average, how many times would you expect to get less than 2 fives? (d) Compare your answers above to the results from your experiment. Comment on accuracy of the predictions from the theoretical model.

5 Probability modeling using the binomial distribution. (Idea from teaching statistics PD from MEI) What s the probability that in this class someone has a mobile phone contaminated with E-coli bacteria of a faecal origin? A study has found that 41 per cent of Britons admit to regularly using their phones, tablets and ereaders when sat on the loo. Smartphones were identified as the preferred gadget of choice, cited by 65 per cent of respondents. Read more: Follow on Twitter DailyMail on Facebook One in six mobile phones contain E coli Some assumptions for discussion The probability of a contaminated phone is independent, like the rolling of dice. The data above is sufficiently reliable to allow an estimate of p (the probability of a phone being contaminated) to be calculated. The phone is either contaminated or not. The number of people in the class is fixed. Card sort introduce or consolidate notation

6 Frequency Data on the distance travelled to work each day (UK Census 2001) Census 2001 data on the average distance travelled to work each day The average distance travelled to work each day (km) Summary Statistics Distance (km) Frequency 6<x 8 3 8<x <x <x <x <x <x <x Average distance in km Raw data Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km) Rank Dist(km)

7 Frequency Data on the October rainfall in Sheffield ( ) Met office figures for October rainfall ( in Sheffield) Summary Statistics Rainfall (mm) Frequency 0<x <x <x <x <x< <x <x <x <x <x October rainfall (mm)

8 Tasks using the data sheets Using the information about the distance travelled to work each day. 1. Find the value of (a) the mean + 1 S.D. (b) the mean 1 S.D. 2. The raw data has been placed in size order. Look at the raw data and find how many of the distances are between the answer to (a) and (b). 3 Work out what percentage of the 348 distances are in the interval mean ± 1 S.D 4 Find the value of (a) the mean + 2 S.D. (b) the mean 2 S.D. 5. Look at the raw data and find how many of the distances are between the answer to (a) and (b). 6. Work out what percentage of the 348 distances are in the interval : (a) mean ± 2 S.D (b) mean ± 3 S.D Using the information about October rainfall in Sheffield. 7. Find the value of (a) the mean + 1 S.D. (b) the mean 1 S.D. 8. The raw data has been placed in size order. Look at the raw data and find how many of the figures are between the answer to (a) and (b). 9. Work out what percentage of the 133 figures are in the interval mean ± 1S.D 10. Find the value of (a) the mean + 2 S.D. (b) the mean 2 S.D. 11. Look at the raw data and find how many of the figures are between the answer to (a) and (b). 12. Work out what percentage of the 133 figures are in the interval : (a) mean ± 2 S.D (b) mean ± 3 S.D 13. Compare your answers to 3 and 6 for both sets of data comment on your observations. Further Reading

Frequency Frequency Data on the distance travelled to work each day (UK Census 2011) The average distance travelled to work each day (km) Summary Statistics Distance (km)

You are a reporter for a national newspaper and you have been asked to write an article which compares the distance people travelled to work in 2001 with the distance

It is suggested you use diagrams and appropriate figures to support your conclusions.

9 Frequency Frequency Data on the distance travelled to work each day (UK Census 2011) The average distance travelled to work each day (km) Summary Statistics Distance (km) Frequency 6<x 8 1 8<x <x <x <x <x <x <x <x <x <x <x <x 32 1 Average distance in km Task. You are a reporter for a national newspaper and you have been asked to write an article which compares the distance people travelled to work in 2001 with the distance people travelled to work in You have been asked to use the Census data for both years. It is suggested you use diagrams and appropriate figures to support your conclusions. Data on the distance travelled to work each day (UK Census 2001) Census 2001 data on the average distance travelled to work each day The average distance travelled to work each day (km) Summary Statistics Distance (km) Frequency 6<x 8 3 8<x <x <x <x <x <x <x Average distance in km

10 Match these distributions to the descriptions on the next page

Orientation and Conferencing Plan Stage 1

Orientation and Conferencing Plan Stage 1 Orientation Ensure that you have read about using the plan in the Program Guide. Book summary Read the following summary to the student. Everyone plays with the