It's great when a different perspective confirms an agreement! I can sleep better.
Linda -----Original Message----- From: programming-boun...@jsoftware.com [mailto:programming-boun...@jsoftware.com] On Behalf Of km Sent: Monday, February 13, 2012 11:10 PM To: Programming forum Subject: Re: [Jprogramming] Challenge 5 Super Bowl Supposition We are in agreement. Your 0.34375 is the probability that B wins, so the probability A wins is 1-0.34375 0.65625 which is what I got. Kip Sent from my iPad On Feb 13, 2012, at 9:07 PM, "Linda Alvord" <lindaalv...@verizon.net> wrote: > Here's my thinking about the theoretical probabilities.s (Here an AFL win > is an 8) ] > > td=: fd (6*2=y)+ y=:, 2+1{."1 I.3=+/\"1 #:i.64 > 4 8 > 5 12 > 6 12 > 7 10 > 8 22 > > ]tpd=:(4+i.5),. ({:"1 td) %+/{:"1 td > 4 0.125 > 5 0.1875 > 6 0.1875 > 7 0.15625 > 8 0.34375 > > I haven't had time to figure out why we disagree. More later. > > Linda > > -----Original Message----- > From: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] On Behalf Of Kip Murray > Sent: Monday, February 13, 2012 2:03 PM > To: Programming forum > Subject: Re: [Jprogramming] Challenge 5 Super Bowl Supposition > > Here is a partial solution to Linda's Challenge 5. --Kip Murray > > Let us call the teams A and B. A has won the first game of a seven game > series, and there are up to six games to go. I deal with the question > of whether A wins the series, ignoring the matter of how many games are > required. Each game is played until there is a winner -- there are no ties. > > Let us imagine the teams play all six remaining games even if the series > is decided before the sixth remaining game. Because the teams are > equally matched, the six remaining games are equivalent to six tosses of > a fair coin with sides 0 and 1, where 1 means a win by team A. A key > idea is that A wins the series if A wins three or more of the remaining > six games, because then A has won four or more games in all and B has > won at most three games in all. > > The binomial distribution (x!y)%(2^y) gives the probability of obtaining > exactly x heads in y tosses of a fair coin. For six tosses the > probabilities are shown in the table > > (] ,: (2^6) %~ 6 !~ ]) i.7 > 0 1 2 3 4 5 6 > 0.015625 0.09375 0.234375 0.3125 0.234375 0.09375 0.015625 > > We find the probability of three or more heads by summing: > > ([: +/ 3 }. (2^6) %~ 6 !~ ]) i.7 > 0.65625 > > That is the theoretical probability that A wins the series, given that A > has won the first game. > > Verb winpct below simulates y repetitions of tossing a fair coin 6 times > and returns the percent of those y repetitions which resulted in 3 or > more heads. (Heads is 1, tails is 0.) > > winpct =: 100 * ] %~ [: +/ 3 <: [: +/ [: ? 2 $~ 6 , ] > > winpct 2000000 > 65.6252 > > winpct"0 [ 5#2000000 > 65.5622 65.6496 65.599 65.6166 65.6298 > > > On 1/31/2012 3:46 AM, Linda Alvord wrote: >> Challenge 5 Super Bowl Supposition PLEASE DO NOT RESPOND UNTIL 2/6/2012 > 12 >> am EST >> >> >> >> As the Super Bowl approaches, suppose it will be decided like baseball. > Four >> of seven games determines a winner. Also suppose that the NFL has won the >> first game. >> >> >> >> Simulate results of 2000000 series and provide the number of times the NFL >> wins in 4 5 6 7 games. If the AFL wins this Extended Super Bowl >> Contest, the result is an 8 . Create a 2000000 item list of number of >> games necessary to determine a winner and provide a frequency > distribution. >> >> >> >> fd=: [: /:~ ({. , #)/.~ >> >> fd (expression for 2000000 trials) >> >> 4 249561 >> 5 374865 >> 6 373851 >> 7 312603 >> 8 689120 >> >> ]games=:fd n,.2000000$6 >> >> 4 249301 >> 5 376266 >> 6 375281 >> 7 311189 >> 8 687963 >> >> ]prob=:(4+i.5),. (1{"1 games)%2000000 >> >> 4 0.124651 >> 5 0.188133 >> 6 0.18764 >> 7 0.155595 >> 8 0.343982 >> >> >> >> ]+/(1{"1 games)%2000000 >> >> 1 >> >> >> >> Now, confirm that your results are reasonable with a theoretical > argument. >> >> >> >> Also, enjoy the Super Bowl! >> >> >> >> Linda >> >> >> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
