Question

In somerset, about 20% of people support legalizing Marijuana. In Morristown, about 60% support it. You conduct the following experiment:

1) You send 1 person to morristown, they count how many people they have to question before 1 agrees that they support the proposition.

2) Send 1 person to somerset, they count how many people they have to question before 1 agrees to support the proposition.

You recieve answers back the next day, 3 and 14. But you don't know which answer goes with which town.

a) For each town, answer combination, calculate the probability of the answer given the town.

b) Assuming 0.5 probability for each town before you look at the answers, what is the probability of Morristown given the answer 14?

Answer 1

Part a)

Answer combination is given as below:

Somerset = 14

Morristown = 3

Now, we have to find the probability that the 14^th person in Somerset will support legalizing Marijuana.

Here, we have to use negative binomial distribution.

We are given X=14, r=1, p = 0.20

Formula is given as below:

P(X=x) = _{(x – 1)}C_{(r – 1)}*p^r *(1 – p)^{(x – r)}

P(X=14) = (14 – 1)C(1 – 1)*0.20^1*(1 – 0.20)^(14 – 1)

P(X=14) = 13C0*0.20^1*0.80^13

P(X=14) = 1*0.20* 0.054975581

P(X=14) = 0.010995116

Required probability = 0.010995116

Now, we have to find the probability that the 3^rd person in Morristown will support legalizing Marijuana.

Here, we have to use negative binomial distribution.

We are given X=3, r=1, p = 0.60

Formula is given as below:

P(X=x) = _{(x – 1)}C_{(r – 1)}*p^r *(1 – p)^{(x – r)}

P(X=3) = (3 – 1)C(1 – 1)*0.60^1*(1 – 0.60)^(3 – 1)

P(X=3) = 2C0*0.60^1*0.40^2

P(X=3) = 1*0.60*0.16

P(X=3) = 0.096

Required probability =0.096

Part b)

We are given X=14, r=1, p = 0.5

Formula is given as below:

P(X=x) = _{(x – 1)}C_{(r – 1)}*p^r *(1 – p)^{(x – r)}

P(X=14) = (14 – 1)C(1 – 1)*0.5^1*(1 – 0.5)^(14 – 1)

P(X=14) = 13C0*0.5*0.5^13

P(X=14) = 1*0.5* 0.00012207

P(X=14) = 0.0000610352

Required probability = 0.0000610352