22​% of a certain​ country's voters think that it is too easy to vote in their country. You ...

For this you use the binomial distribution formula: \(P\left(r\right)=_nC_r\times \pi ^r\left(1-\pi \right)^{n-r}\), where \(\pi\) is the probability.

22% is equivalent to 0.22 in decimal form.

a) For the probability of exactly 3, you set n = 12, r = 3, \(\pi\) = 0.22, and \(1-\pi =0.78\)

\(P\left(3\right)=_{12}C_3\times 0.22^3\left(0.78\right)^{12-3}\)

\(P\left(3\right)=_{12}C_3\times 0.22^3\left(0.78\right)^{12-3}=220\times 0.00113=0.250\)

b) For this, you have the option of finding the probability of 4 all the way to 12, or finding the sum of the probabilities of 0 to 3, and subtracting it from 1. I'll use the latter method:

\(P\left(r\ge 4\right)=1-\left[P\left(3\right)+P\left(2\right)+P\left(1\right)+P\left(0\right)\right]\)

\(1-\left[_{12}C_3\cdot 0.22^3\left(0.78\right)^9+_{12}C_2\cdot 0.22^2\left(0.78\right)^{10}+_{12}C_1\cdot 0.22^1\left(0.78\right)^{11}+_{12}C_0\cdot 0.22^0\left(0.78\right)^{12}\right]\)

\(P\left(r\ge 4\right)=0.261\)

c) For less than 8, find the probabilities of 8 to 12, then subtract from 1.

\(P\left(r<8\right)=1-\left[P\left(8\right)+P\left(9\right)+P\left(10\right)+P\left(11\right)+P\left(12\right)\right]\)

\(1-\left[_{12}C_8\cdot 0.22^8\left(0.78\right)^4+_{12}C_9\cdot 0.22^9\left(0.78\right)^3+_{12}C_{10}\cdot 0.22^{10}\left(0.78\right)^2+_{12}C_{11}\cdot 0.22^{11}\left(0.78\right)^1+_{12}C_{12}\cdot 0.22^{12}\left(0.78\right)^0\right]\)

\(P\left(r<8\right)=0.9989\rightarrow 0.999\)