P(at least two students sharing the same birthday)

Transcription

1 () Itoductio Bithday aadox Yue Kwok hoy lease make a guess: I a class of 3 studets, what is the pobability that at least two studets shaig a bithday, assumig that thee ae days i a yea (foget about the leap yea)? This is the classical mathematical poblem called the bithday paadox. I fact, this is ot eally a paadox, as paadoxes usually lead to logical cotadictios. We call this a paadox because oe ot takig pope pobability taiig teds to ude-estimatig the value of this pobability. He may have a expeiece that it is difficult fo him to fid aothe oe i a small goup (such as 3 hee) to have the same bithday as his. Howeve, we ae ot demadig the coicidece betwee specific studets ad specific bithdays, but just ay two studets ad ay bithday i a yea. The esultig pobability is theefoe highe. The calculatio is ot difficult. If o two shae a bithday, the the fist studet ca choose a bithday i ways, the secod studet i 364 ways, the thid i 363 ways, ad so o. If thee ae 3 studets i the oom, the pobability that o two shaig a bithday is: (oe shaig the same bithday) (at least two studets shaig the same bithday) () 3 Thee is just moe tha half the pobability that at least two studets havig the same bithday i a class of just 3 studets. () Factoial ad emutatio If we have studets i a class istead of 3, the (at least two studets shaig the same bithday) () If we defie, fo ay atual umbe, the factoial of :! 3 ad 0! ( )( ) the we all kow that! is the umbe of ways to aage diffeet objects i a sequece.

2 Equatio () ca be ewitte i the fom: (at least two studets shaig the same bithday)! (3) ( )! If we futhe defie, fo o-egative iteges ad, 0,! the pemutatio: ( )! the we kow that gives us the umbe of ways i choosig objects out of diffeet objects ad aagig these objects i a sequece i which ode is impotat. Equatio (3) ca be futhe ewitte i the fom: (at least two studets shaig the same bithday) (4) (3) ombiatio How ca we kow that 3 studets ae eough to get a pobability of just ove 0.5 with at least studets shaig the same bithday? Oe method is to use a compute pogam ad fid the coespodig pobability fo, 3, 4,. owads util you ca get a value of pobability moe tha 0.5. A small EXEL speadsheet is attached fo this pupose. ROBABILITY AT LEAST ERSONS SHARING THE SAME BIRTHDAY obability Numbe of people

3 Aothe method which is moe mathematical is to solve fo a smallest itege, > 0.5 o 0. 5 Howeve, this is difficult to solve. < o < 0.5( ) So if we defie, fo o-egative iteges ad, 0,! the combiatio :! ( )! the we kow that gives us the umbe of ways i choosig objects out of diffeet objects ad aagig these objects i a sequece i which ode is uimpotat. 364 The pobability of ay two studets ot havig the same bithday is. I a class ( ) cotaiig studets, thee ae pais of studets. The pobability of o two studets shaig the same bithday ca be appoximated by assumig that these evets ae idepedet ad hece by multiplyig thei pobability togethe. Theefoe we have 364 (at least two studets shaig the same bithday) (5) It is eally a appoximatio as the pais (A, B) ( ) ad (B, ) havig commo bithdays does ot imply that (A, ) have o commo bithday. Howeve, the pobability i (5) is a good appoximatio if is gettig bigge ad the umbe of pais is eve bigge. So if we wat to fid the smallest positive itege to have a pobability of 0.5 we solve: 364 ( ) > 0.5 o Takig logaithm, we get : ( ) log > log o 364 ( ) 364 ( ) < 0.5 o ( ) 364 > < log / log Hece, we eed to solve the quadatic iequality: < 0 (6) Solvig (6), usig quadatic equatio fomula to half, we get: < o > (7) Sice is the smallest positive itege satisfyig (7), we get 3. 3

4 (4) Thee studets shaig bithday It is athe difficult to calculate the pobability that at least 3 studets shaig a bithday i a class of studets. We fid out the pobability that o 3 studets shaig a bithday fist ad the emaide is what we wat. (a) Ω {all possible combiatio fo studets} N(Ω) (b) A {oe of studets havig the same bithday} N(A) (c) (i) B { pai of studets ( studets) havig the same bithday} N(B ) (ii) B { pais of studets (i.e. 4 studets) havig the same bithday} N(B )! (iii) B 3 {3 pais of studets (i.e. 6 studets) havig the same bithday} 4 3 N(B ) 3! (iv) If we cotiue i this way we get the last possible umbe of pais: B / { / pais of studets (i.e. / studets) havig the same bithday} whee / stads fo the lagest possible itege smalle tha /. (v) N(B / ) 4 / / /! B {oly pais studets havig the same bithday} By summig up all the possible pais listed i (c) (i) to (iv), we get: N(B)!! i i i! i i! + / 4 (i ) / i ( i)! (8) (d) {at least 3 studets havig the same bithday} () (A) (B) / i 4 i! (i ) i / i i i!!! ( + i)! (9) 4

5 (5) Multiomial oefficiet The multiomial coefficiet is a geealizatio of the biomial coefficiet. Let k, k,., k m be o-egative iteges givig a patitio of :! k + k +.+ k m. The multiomial coefficiet is defied by k!k!...k! m. This coefficiet gives you the umbe of pemutatios of objects, i which k ae alike, k ae alike,., k m ae alike. The bithday paadox ca the be solved with the help of multiomial theoem usig mables-i-jas model (MJ model). (Nothig coected with Michael Jackso hee. ) Fist we defie : x : y,,y as thee ae x jas ad thee ae y alike mables iside each ja. So suppose we like to fid the pobability of havig exactly studets with the same bithdays ad the est of the studets ae of diffeet bithdays i a class of 3 studets. The coespodig MJ model is : :, :,,, 343 : 0,,0 (Itepetatio : ja with alike mables, jas with mables, 343 jas with o mable.) alculatio of the pobability ivolves 3 steps: () alculatio of the umbe of ways i aagig the Jas, hee we have: J!!! 343! () alculatio of the umbe of ways i aagig the Mables i the jas, hee : 3! M!! 0! ( ) ( ) 343 (3) Fid MJ ad divide by the total umbe of ways of aagig 3 mables i jas, that is 3, hee fially we get : (oe ad oly oe pai of same bithday)! 3! (! ) ( 0! )!!! 343! (0) This pobability is of couse smalle tha the pobability listed i (). 5

6 (6) Moe calculatios with MJ model (a) obability of o studets with same bithday i a class of : MJ model : :,,, 35 : 0,,0 Remembe that obability MJ / N(Ω) (o studets with same bithday)! (! ) ( 0! )!! 35! Also, we ca calculate: (at least two studets shaig the same bithday i a class of ) (Quite high!) (b) obability of exactly two pais of studets shaig same bithday i a class of : MJ model : :,, 36 :,,, 37 : 0,,0 (exactly two pais of studets shaig same bithday)! 36 (! ) (! ) ( 0! )!! 36! 37! (c) obability of exactly thee studets shaig same bithday i a class of : MJ model : : 3, 37 :,,, 37 : 0,,0 (exactly 3 studets with same bithday, o pai shaig same bithday)! 37 ( 3! )(! ) ( 0! )!! 37! 37! Hade Execise () Use the MJ model to wok out (9). () Use EXEL o ay softwae to make a gaph fo (9) fo 3, 4,, 00. Sed to M. Yue if you like. 6