Question

Profile of a sustainable farmer. Sustainable development or sustainable farming means finding ways to live and workthe Earth without jeopardizing the future. Studies wereconducted in five Midwestern states to develop a profileof a sustainable farmer. The results revealed that farmerscan be classified along a sustainability scale, dependingon whether they are likely (L) or unlikely (U) to engagein the following practices: (1) raise a broad mix of crops;(2) raise livestock; (3) use chemicals sparingly; and (4) usetechniques for regenerating the soil, such as crop rotation.

1. List the different sets of classifications that are possible for the four practices (e.g., LUUL).
2. Suppose you plan to interview farmers across the country to determine the frequency with which they fall into the classification sets you listed for part a. Because no information is yet available, assume initially that there is an equal chance of a farmer falling into any single classification set. Using that assumption, what is the probability that a farmer will be classified as unlikely on all four criteria (i.e., classified as a non-sustainable farmer)?
3. Using the same assumption as in part b, what is the probability that a farmer will be classified as likely on at least three of the criteria (i.e., classified as a near sustainable farmer)?

Short answer

1. The samples are LLLL, LLLU, LLUL, LULL, ULLL, LLUU, LULU, LUUL, ULLU, ULUL, UULL, LUUU, ULUU, UULU, UUUL, UUUU
2. The probability of unlikely on all four criteria is $\frac{\mathbf{1}}{16}$.
3. The probability is likely on at least three of the criteria.

Step-by-step solution

Important formula

The formula for probability is $\mathbf{P}\mathbf{=}\frac{\mathrm{Favourable}\mathrm{out}\mathrm{comes}}{\mathrm{Total}\mathrm{out}\mathrm{comes}}$

(a) List the different sets of classifications that are possible for the four practices

L = likely

U = unlikely

The different sets of classifications are:

LLLL, LLLU, LLUL, LULL, ULLL, LLUU, LULU, LUUL, ULLU, ULUL, UULL, LUUU, ULUU, UULU, UUUL, UUUU.

So, the total events are 16.

(b) Find the probability that a farmer will be classified as unlikely on all four criteria

The sample as unlikely on all four criteria is UUUU.

$\mathbf{P}\mathbf{\left(}\mathbf{UUUU}\mathbf{\right)}\mathbf{=}\frac{1}{16}$.

Hence, the probability of unlikely on all four criteria is $\frac{1}{16}$.

(c) Determine the probability that a farmer will be classified as likely on at least three of the criteria

The sample is likely based on at least three criteria: LLLL, LLLU, ULLL, LLUL, and LULL.

So,$\mathbf{P}\mathbf{\left(}\mathbf{UUUU}\mathbf{\right)}\mathbf{=}\frac{5}{16}$ .

Therefore, the probability is likely on at least three of the criteria.