if 60% of the electorate supports the mayor what is the probability that in a random sample of 10 voters, fewer than half support her

Question

 I just really need help with this question. I was not in class when we learned about this. 

Answer 1

Let 0.6 be the percentage that support the mayor, then 0.4 do not support the mayor.

 

If fewer than half support the mayor, that means either 0, 1, 2, 3, 4 support the mayor

 

Let k = 0, 1, 2, 3, 4.  Then for each of there are  (10 C k) = 10!/(k! * (10 - k)!) distinct ways the values in the random sample can be determined so that k support the mayor.

 

p₀ = probability that no on supports the mayor =  (10 C 0) * .6⁰ * .4¹⁰ = 1 *  .6⁰ * .4¹⁰ 

p₁ = probability that only1 person supports the mayor = (10 C 1)  * .6¹ * .4⁹ = 10 * .6¹ * .4⁹

p₂ = probability that only 2 persons support the mayor = (10 C 2) * .6² * .4^₈ = 45 * .6² * .4⁸

p₃ = probability that only 3 persons support the mayor = (10 C 3) * .6³ * .4⁷ =120 *  .6³ * .4⁷

p₄ = probability that only 4 persons support the mayor = (10 C 4) * .6⁴ * .4⁶ = 210 *  .6⁴ * .4⁶

 

∴The probability that fewer than half the sample support the mayor is p₀ + p₁ + p₂ + p₃ + p₄ ≈ 0.1662