Question

Use Scenario 6-16. If the reporter asks adults on the street at random, what is the probability that he will find a Democrat by the time he has stopped three people?

Answer 1

`P(X\leq 3) = P(X=1) +P(X=2) +P(X=3)`

`P(X=1) = (1-0.6)^(3-1) 0.6 = 0.096`

`P(X=2) = (1-0.6)^(3-2) 0.6 = 0.24`

`P(X=3) = (1-0.6)^(3-3) 0.6 = 0.60`

`P(X\leq 3) = 0.096 +0.24 +0.60= 0.936`

Explanation:

Assuming the following info for the scenario 6-16: "Scenario 6-16

A poll shows that 60% of the adults in a large town are registered Democrats. A newspaper reporter wants to interview a local democrat regarding a recent decision by the City Council. "

Previous concepts

The geometric distribution "represents the number of failures before you get a success in a series of Bernoulli trials". And the density function is given by:

`P(X) = (1-p) ^(x-1) p`

Where
`X \geq 1`

what is the probability that he will find a Democrat by the time he has stopped three people?

For this case we can assum that the random variable X represent the number of people selected in order to find a democrat and for this case we can assume that our random variable follows a geometric distribution.

`X \sim Geome (p=0.60)`

We can find a democrat selecting 1, 2 or 3 people so we need to find this probability.

And for this case we want this probability:

`P(X\leq 3) = P(X=1) +P(X=2) +P(X=3)`

`P(X=1) = (1-0.6)^(3-1) 0.6 = 0.096`

`P(X=2) = (1-0.6)^(3-2) 0.6 = 0.24`

`P(X=3) = (1-0.6)^(3-3) 0.6 = 0.60`

`P(X\leq 3) = 0.096 +0.24 +0.60= 0.936`