Probablty In ths actvty you wll use some real data to estmate the probablty of an event happenng. You wll also use a varety of methods to work out theoretcal probabltes. heoretcal and expermental probabltes can be used to predct the number of trals n whch an event wll occur. Informaton sheet Relatve frequency Suppose you want to fnd the probablty of a drawng pn landng pont up when you toss t up n the ar. o estmate the probablty you could carry out an experment to fnd n what fracton of the trals the drawng pn lands pont up. hs fracton, called the relatve frequency, gves an estmate of the probablty. For example, f the drawng pn was tossed 00 tmes and t landed pont up on 40 of these trals, P(drawng pn lands pont up) 40 7 = 0.7 00 0 he greater the amount of data used, the more relable the relatve frequency s as an estmate of the probablty. For example, dong the experment 000 tmes would gve a better estmate of the probablty than dong t 00 tmes. In other stuatons you mght use a survey or data that has been collected n the past. For example, f you wanted to estmate the probablty of a tran beng late you could look at the tran operator s past record on punctualty for that journey. Agan, the more data used, the better the estmate. hnk about Can you thnk of other stuatons where data can help you to estmate probabltes? heoretcal probablty When somethng can happen n a number of equally lkely ways, the probablty of event, E, can be calculated usng: For example, f a far dce s thrown, there are possbltes.... Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page of 7
P() hs means the probablty of gettng a sx s P(evennumber) P(or4or) P(morethan) hnk about P(or4or5or) ow do these examples use the fact that the dce s far? he probablty of somethng that s mpossble s 0. For example, on the dce P(7) = 0. he probablty of somethng that s certan s. For example, on the dce P(score < 7) =. hese results can all be shown on a probablty lne: 4 0 P(7) P() P(even) P(more than ) P(less than 7) E and not E As t s certan that an event E ether occurs or does not occur P(E doesn t happen) + P(E happens) =. Usng the notaton P(E) for the probablty that E doesn t happen, P( E) P(E). hnk about Use ths result and the probabltes gven above to answer these questons: What s the probablty of the drawng pn from the frst example not landng pont up? For a far dce, what s the probablty of not gettng a? What s the probablty of gettng at most when a far dce s thrown? Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page of 7
Expectaton Probabltes can be used to make predctons. For example, f a far dce s thrown 00 tmes, the expected number of sxes s one sxth of 00, whch s 50. In general: Expected number of trals n whch E occurs = P(E) total number of trals Usng the probabltes gven earler, expected number of even scores = 00 = 50 expected number of scores above = 00 = 00 Combned events If two far cons are tossed the possbltes are: hese can be systematcally lsted, as above, or shown n a table, or n a tree dagram. nd con st con st con nd con Each of these methods gves all four possble outcomes, and allows probabltes to be found. For example: s one of the four equally lkely outcomes, so the probablty of gettng heads s 4. Smlarly, the probablty of gettng a head and tal n any order = 4 = he probablty of gettng the same on each con = 4 = he probablty of gettng at least one tal = 4 hnk about Can you use P( E) P(E) to relate these results? Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page of 7
ry these Lost luggage he table gves the number of bags checked n for Brtsh Arways and bm flghts n 007 and the number of these bags that were delayed or lost. Number of bags checked n delayed/lost Brtsh Arways 4 04 4 4 05 bm 5 9 50 88 90 a Use ths data to estmate the probablty that: a bag checked n for a Brtsh Arways fght wll be delayed or lost a bag checked n for a bm fght wll be delayed or lost b If Brtsh Arways do not delver a bag at the end of a flght, the probablty that t wll be returned wthn 48 hours s 0.85. What s the probablty that a delayed bag wll not be returned wthn 48 hours? 450 bags are checked n for a Brtsh Arways flght. ow many of these bags do you expect to be delayed? ow many of the bags do you expect to be delayed more than 48 hours? Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page 4 of 7
Dce dfferences a he table shows the possble dfferences between the scores of two far dce when they are thrown. One row of the table has been flled n. Complete the other rows. st dce nd dce 4 5 0 4 4 5 b Complete the table to gve the probablty of each of the possble dfferences. Dfference 0 4 5 Probablty c What s the probablty that the dfference s: greater than? less than or equal to?... rd con hree cons a Complete the dagram to show all the possbltes when far cons are tossed. nd con b What s the probablty that all cons show tals? st con... all cons show the same? there are heads and tal?.... Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page 5 of 7
4 Chldren a Each chld could be a boy or grl. Lst all the possbltes for chldren....... b Assumng that any chld s equally lkely to be a boy or grl, fnd the probablty that f a couple have chldren: all the chldren wll have the same gender two of the chldren wll be boys. there wll be more boys than grls. c Lst all the possbltes for 4 chldren............. d Assumng that any chld s equally lkely to be a boy or grl, fnd the probablty that a couple who have 4 chldren have an equal number of boys and grls.... Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page of 7
Extenson What connectons are there between probabltes for the three con tosses n Queston and those for the three chldren n Queston 4? Why? Queston s about the dfferences n the scores when two far dce are thrown. Make up and solve a probablty queston when the scores on two dce are combned n some other way, such as addng or multplyng, or takng the larger score. Reflect on your work Explan how relatve frequency s used to estmate probabltes. Gve some examples. Wrte down a formula for the theoretcal probablty of an event E. Why s t mportant to assume that dce and cons are far when you are calculatng theoretcal probabltes? Descrbe three ways of fndng probabltes of combned events. Nuffeld Free-Standng Mathematcs Actvty Probablty Student sheets Copable page 7 of 7