In a certain British soccer pool,
the objective is to pick games
that end in a tie. The ticket buyer
picks 8 games from a list of 45 or
more. For each of these games,
if the teams tie, the player gets 3
points, if the visiting team wins,
they get 2 points, and if the home
team wins, they get 1.5 points. The
entry with the highest point total
wins.
Assume that for each game,
the probability of the home team’s
winning is 0.5, the probability of
the visiting team’s winning is 0.4,
and the probability of a tie is 0.1.
Determine the probability that the total
points for an entry will be 22 or
higher.
clearvars,clc
p=sym(zeros(1,48));
p(3)=1/2;p(4)=2/5;p(6)=1/10;
for game=2:8
newp=sym(zeros(1,48));
for pPos=1:48
if p(pPos)>0
newp(pPos+3)=newp(pPos+3)+p(pPos)/2;
newp(pPos+4)=newp(pPos+4)+p(pPos)*2/5;
newp(pPos+6)=newp(pPos+6)+p(pPos)/10;
end
end
p=newp;
end
for i=1:length(p)
if p(i)>0
fprintf('%3.1f %16s %10.8f\n',i/2,p(i),eval(p(i)))
end
end
fprintf('%s %10.8f\n',sum(p(44:48)),eval(sum(p(44:48))))
finds the probabilities of each possible total:
total approximate
pts. probability decimal
12.0 1/256 0.00390625
12.5 1/40 0.02500000
13.0 7/100 0.07000000
13.5 473/4000 0.11825000
14.0 147/1000 0.14700000
14.5 973/6250 0.15568000
15.0 145047/1000000 0.14504700
15.5 73221/625000 0.11715360
16.0 133849/1562500 0.08566336
16.5 145047/2500000 0.05801880
17.0 109971/3125000 0.03519072
17.5 1547/78125 0.01980160
18.0 527219/50000000 0.01054438
18.5 623/125000 0.00498400
19.0 3521/1562500 0.00225344
19.5 483/500000 0.00096600
20.0 217/625000 0.00034720
20.5 21/156250 0.00013440
21.0 1071/25000000 0.00004284
21.5 7/625000 0.00001120
22.0 7/1562500 0.00000448
22.5 1/2500000 0.00000040
23.0 1/3125000 0.00000032
24.0 1/100000000 0.00000001
The total probability of 22 points or higher:
521/100000000 = 0.00000521
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Posted by Charlie
on 2024-12-08 08:25:50 |
computer-aided solution
