In the game “Drop Dead,” you roll
a number of six-sided dice. If a roll
does not include any 2s or 5s, you
add the sum of the dice to your
score and roll all of the dice again.
If your roll does include 2s or 5s,
you receive no points for that roll,
the dice with 2s or 5s are discarded,
and the remaining dice are rolled again. You repeat this procedure until all dice have been discarded.
If you start with five dice, what is your
expected score by the time you have
discarded all of your dice?
(In reply to
Got it. Program based on Analytic formula by Larry)
I translated the program into Matlab language so I could repeat the calculation in rational form:
clearvars,clc
s_of_n = 0;
for n =1:11
multiplier = (3^n) / (3^n - 2^n);
answ = (2/3)^n * 3.5 * n;
for k =1:n
answ=answ+ (nchoosek(n,k) ...
* (1/3)^k ...
* (2/3)^(n-k) ...
* s_of_n(n-k+1));
end
s_of_n(end+1)=answ*multiplier;
end
for i=1:length(s_of_n)-1
disp([i,s_of_n(i+1)])
end
clearvars
s_of_n = sym(0);
for n =sym(1):11
multiplier = (3^n) / (3^n - 2^n);
answ = (2/3)^n * 3.5 * n;
for k =1:n
answ=answ+ (nchoosek(n,k) ...
* (1/3)^k ...
* (2/3)^(n-k) ...
* s_of_n(n-k+1));
end
s_of_n(end+1)=answ*multiplier;
end
for i=1:length(s_of_n)-1
disp([i,s_of_n(i+1)])
end
1 7
2 11.2
3 13.7052631578947
4 15.1902834008097
5 16.0646622023524
6 16.5758274651265
7 16.8723924686845
8 17.0430710032829
9 17.1404691835483
10 17.1955586468927
11 17.2264332639042
[1, 7]
[2, 56/5]
[3, 1302/95]
[4, 3752/247]
[5, 837242/52117]
[6, 4319412/260585]
[7, 9052789634/536544515]
[8, 177400705552/10408963591]
[9, 180020250341190/10502644263319]
[10, 9932935945754444/577645434482545]
[11, 158397269869263071038/9195012539314467905]
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Posted by Charlie
on 2024-12-03 16:00:46 |
re: Got it. Program based on Analytic formula
