PROBABILITY Probability from data tables and frequency trees

Probability from data tables In GCSE Statistics you will be asked to give estimates of probability and expected frequencies from a data set. Dwellings Dwelling Type: Bungalow/Detac hed Area Northern Ireland Dwelling Type: Bungalow/Detac hed (%) Dwelling Type: Terraced/Semi. Detached House (%) Dwelling Type: Flat/Apartment (%) 760000 318530 41. 9 374990 49. 3 66480 8. 7 Antrim and Newtownabbey 60060 22020 36. 7 26840 44. 7 11200 18. 6 Ards and North Down 70960 31180 43. 9 31980 45. 1 7800 11 80350 35710 44. 5 40640 50. 6 4000 5 123590 15830 12. 8 88610 71. 7 19150 15. 5 Causeway Coast and Glens 60900 33830 55. 6 21990 36. 1 5080 8. 3 Derry City and Strabane 59240 26230 44. 3 30110 50. 8 2900 4. 9 Fermanagh and Omagh 54350 34160 62. 8 16730 30. 8 3460 6. 4 Lisburn and Castlereagh 69640 20500 29. 4 45040 64. 7 4100 5. 9 Mid and East Antrim 62480 29110 46. 6 27740 44. 4 5630 9 52150 32360 62 19160 36. 7 630 1. 2 66280 37600 56. 7 26150 39. 5 2530 3. 8 Armagh City, Banbridge and Craigavon Belfast Mid Ulster Newry, Mourne and Down

The data set on the previous slide shows the number of dwellings by area in Northern Ireland in the 2011 census. • a) Estimate the probability that a dwelling in Northern Ireland in 2011 is a flat or apartment? • b) Estimate the probability that a dwelling in Northern Ireland in 2011 was situated in Mid Ulster? • c) Why are the answers for a) and b) estimates? • d) If 500 properties were picked at random in Belfast how many of them would you expect to be Bungalow/Detached houses?

Solution • a) The percentage of flats and apartments in Northern Ireland is 8. 7%. As a probability this would be given as 0. 087 • b) There were 52150 dwellings in Mid Ulster out of a total of 760000. • The probability is 52150 ÷ 760000 = 0. 069 (2 sf) • c) The probabilities are estimates as the total number of dwellings has been rounded. • d) The probability of a property in Belfast being Detached house or Bungalow is 12. 8% or 0. 128 • The expected number of Bungalow/Detached out of 500 = 0. 128 x 500 = 64

Frequency Trees and Tree Diagrams • Frequency trees and tree diagrams are methods of organising information in order to answer probability problems. Both diagrams have a similar layout but with some important differences. • A frequency tree has the data type along the branches and the frequency of the data type at the end of the branch. • A tree diagram has the event at the end of the branch and the probability of the event along the branch. The probabilities in each set of braches must add up to one. We can use the AND rule to multiply probabilities as we travel along the branches.

Example 1 • There are 134 clubs in the Scottish and English football leagues combined. 42 of the clubs are in Scotland. 24 of the clubs in England have won the top division in the country at some time in their history. Only 9 clubs in Scotland have won the top division. • a) Complete a frequency tree to display the information. • b) A team is chosen at random from the English League. What is the probability the team has not won the top division? • c) A team is chosen at random from all of those who have won either the English or Scottish top divisions. What is the probability the team is English?

Solution b) There are 92 teams in England 68 of these have not won the league. The probability is 68/92 c) There are 33 league winners (9 Scottish and 24 English) The probability is 24/33