Question

1. Each year, a town awards 10 grants to support local farms. These are supposed to be decided randomly (equally likely). There are 50 farms with 17 of them being pig farmers and the person running the decision process is a brother to the Pig Farmer’s Association President. Out of the past 10 years, there have been 6 years in which at least 5 pig farms or more have been awarded a grant. Let’s explore how likely that this is from an unbiased random process.

2. 
   

1. What is the probability of 5 pig farms being given the grant in a given year?

2. What is the probability of at least 5 pig farms being given the grant in a given year?

3. What is the probability of exactly 6 years in which at least 5 pig farms have been given a grant?

4. What is the probability of at least 6 years in which at least 5 pig farms have been given a grant?

5. Do you think this process was random or not?

Answer 1

There are 50 farms with 17 of them being pig farmers. The probability that a randomly elected farm is a pig farm is

a) 10 grants are given each year. Let X be the number pig farms given the grant in a given year. We can say that X has Binomial distribution with parameters, number of trials (number of grants awarded) n=10, success probability (  The probability that a randomly elected farm is a pig farm) p=0.34

The probability that X=x pig farms are given the grant in a given year is

the probability of 5 pig farms being given the grant in a given year is

ans: the probability of 5 pig farms being given the grant in a given year is 0.1434

b) the probability of at least 5 pig farms being given the grant in a given year is

ans: the probability of at least 5 pig farms being given the grant in a given year is 0.2270

c.  the probability of at least 5 pig farms being given the grant in a given year is 0.2270. Let Y be the number of years in which  at least 5 pig farms have been given a grant. We can say that Y has a Binomial distribution with parameters, number of trials (total number of years) n=10, success probability (the probability of at least 5 pig farms being given the grant in a given year) p=0.2270

The probability that Y=y years in which  at least 5 pig farms have been given a grant is

the probability of exactly 6 years in which at least 5 pig farms have been given a grant is

ans: the probability of exactly 6 years in which at least 5 pig farms have been given a grant is 0.0103

d. the probability of at least 6 years in which at least 5 pig farms have been given a grant is

ans: the probability of at least 6 years in which at least 5 pig farms have been given a grant is 0.0122

e. Out of the past 10 years, there have been 6 years in which at least 5 pig farms or more have been awarded a grant. If this were from an unbiased random process, the probability of this happening if the is only 0.0103

In fact the probability of at least 6 years in which at least 5 pig farms have been given a grant is 0.0122. That means if this were from an unbiased random process, there is less than 5% chance (to be precise, there is only 1.22% chance ) of at least 6 years in which at least 5 pig farms or more have been awarded a grant.

This is too low to be a result of chance.

Hence we can say that we are 95% confident that this process was not random.