(1) The probability of exactly 25/40 being pro-choice on abortion, if subjected to binomial distribution, that is, (p+q)ⁿ where n=40, p=0.57 and q=1-p=0.43, is given by ⁴⁰C₂₅p²⁵q¹⁵=0.10 or 10% approximately. In other words all arrangements of 40 people such that 25 are pro-choice and 15 are con-choice on abortion.
(2) We need the cumulative probability this time for n=30, so this means summing the relevant terms of the binomial expansion corresponding to (0 pro, 30 con), (1 pro, 29 con), (2 pro, 28 con), ... (15 pro, 15 con). The summation will be q³⁰+30q²⁹p+(30×29/2)q²⁸p²+...+³⁰C₁₆q¹⁶p¹⁴+³⁰C₁₅q¹⁵p¹⁵. By symmetry, ³⁰C₁₆=³⁰C₁₄, etc. The sum comes to 0.276 or about 27.6%. The result can also be obtained from binomial distribution tables, or by using a binomial distribution calculator (also available online).
(3) For this we subtract p(x<20) from p(x≤30) (cumulative binomial probabilities) where n=50.
p(x≤30)=0.71433 and p(x<20)=0.00527, so p(20≤x≤30)=0.7091 or 70.91% approx.
Another way of solving this is to sum the terms from ⁵⁰C₂₀p²⁰q³⁰ to ⁵⁰C₃₀p³⁰q²⁰.