how can i solve equation with summation

i can't solve the attached img equation on matlab  for :  Pfa=10^(-6),M=8,i=s=1
(later i would change these variables)

i tried the following but errors rises

if true
  M=8;
s=1;
i=s;
syms pfal M i s
eqn = symsum((nchoosek(M,i)*(pfal)^(i)*(1-pfal)^(M-i)), i, s, M)==8*10^(-6);
solve(eqn,pfal)   
end

Answer 1

[Rik]

I think I can read the summation index: it is an i. That would mean that you should initiate i as a sym not as a hard-coded value, and thus the result from nchoosek changes each summation part.

M=8;
s=1;
syms pfal i
f = nchoosek(M,i);
eqn = (symsum(f*(pfal)^(i)*(1-pfal)^(M-i), i, s, M)==10^(-6));
solve(eqn,pfal)

This exits without error, but results in 8 solutions, 6 of which are complex. One solution is indeed equal to the solution that Roger suggested (1.25e-7).

[Fatma Abdullah]

2 Thank you so much...that seemed be the problem,,,,could you please tell me how to choose the real less than one solution

[Rik]

Sure. You have to convert the results from solve to another data type, like double. To filter out the non-real solutions, I account for any weird rounding errors by checking if the complex part is smaller than the float precision.

result=solve(eqn,pfal);
result=result( result<1 & abs(imag(result))<2*eps );

[Fatma Abdullah]

sorry to bother but i got that weird error:

Error using mupadmex Error in MuPAD command: Unexpected ']'. [line 1, col 44]

Error in sym/subsref (line 1577) B = mupadmex('symobj::subsref',A.s,inds{:});

Error in OSSOCFAR (line 11) result=result( result<1 & abs(imag(result))<2*eps );

[Rik]

Ah, my bad, I forgot to add the line that converts to double.

result=solve(eqn,pfal);
result=double(result);
result=result( result<1 & abs(imag(result))<2*eps );

Answer 2

[Roger Stafford]

If I interpret your code correctly, you are doing things the hard way. Your expression for Pfa is simply the binomial expansion of ((Pfa1+(1-Pfa1))^M without the first term where i is equal to zero. Hence, you are trying to solve

(Pfa1 + (1-Pfa1))^8 = 1 = 10^(-6) + (1-Pfa1)^8

whose solution is easily found as

Pfa1 = 1 - 0.999999^(1/8)

Probably the cause of your error message is the reference to the numeric function ‘nchoosek’.