Question

if we choose 7 people at random from a population that has 60% democrats, and let x = the number of democrats chosen (out of the 7), then the probability table for x is as follows:
a) what is the probability that less than 6 are democrats?
b) what is the probability that more than 6 are democrats?
c) what is the probability that at least 4 are democrats?

Explanation:

Step1: Calculate probability for part A

Less than 6 means \(P(X < 6)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)+P(X = 5)\)
\(P(X < 6)=0.0016 + 0.0172+0.0774 + 0.1935+0.2903+0.2613 = 0.8413\)

Step2: Calculate probability for part B

More than 6 means \(P(X>6)=P(X = 7)\)
\(P(X>6)=0.028\)

Step3: Calculate probability for part C

At least 4 means \(P(X\geq4)=P(X = 4)+P(X = 5)+P(X = 6)+P(X = 7)\)
\(P(X\geq4)=0.2903+0.2613 + 0.1306+0.028=0.7102\)

Answer:

A. \(0.8413\)
B. \(0.028\)
C. \(0.7102\)