Question

Suppose that in your city 39​% of the voters are registered as​ Democrats, 26​% as​ Republicans, and 7​% as members of other parties. Voters not aligned with any official party are termed​ "Independent." You are conducting a poll by calling registered voters at random. In your first three​ calls, what is the probability that you talk to

​a) all​ Republicans?

​b) no​ Democrats?

​c) at least one​ Independent?

Answer 1

a) 0.0176 or 1.76%

b) 0.2270 or 22.70%

c) 0.6268 or 62.68%

Explanation:

Democrats: P(D) = 39%

Republicans: P(R) = 26%

Others: P(O) = 7%

Independent: P(I) = 100-39-26-7 =28%

a) Probability of talking to all​ Republicans in three calls:

`P(N_R=3) =P(R)*P(R)*P(R)\\P(N_R=3) =0.26^3 = 0.0176`

a) Probability of talking to no democrats in three calls:

`P(N_D=0) =(1-P(D)*(1-P(D)*(1-P(D)\\P(N_D=0) =(1-0.39)^3 = 0.2270`

c) Probability of talking to at least one independent in three calls:

`P(N_I\geq1)=1-P(N_I=0)\\P(N_I\geq1) = 1-(1-0.28)*(1-0.28)*(1-0.28)\\P(N_I\geq1) = 0.6268`