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In the western United States, there are many dry land wheat farms that depend on winter snow and spring rain to produce good crops. About \(65 \%\) of the years there is enough moisture to produce a good wheat crop, depending on the region (Reference: Agricultural Statistics, United States Department of Agriculture). (a) Let \(r\) be a random variable that represents the number of good wheat crops in \(n=8\) years. Suppose the Zimmer farm has reason to believe that at least 4 out of 8 years will be good. However, they need at least 6 good years out of 8 years to survive financially. Compute the probability that the Zimmers will get at least 6 good years out of 8, given what they believe is true; that is, compute \(P(6 \leq r \mid 4 \leq r) .\) See part \((\mathrm{d})\) for a hint. (b) Let \(r\) be a random variable that represents the number of good wheat crops in \(n=10\) years. Suppose the Montoya farm has reason to believe that at least 6 out of 10 years will be good. However, they need at least 8 good years out of 10 years to survive financially. Compute the probability that the Montoyas will get at least 8 good years out of 10, given what they believe is true; that is, compute \(P(8 \leq r \mid 6 \leq r)\). (c) List at least three other areas besides agriculture to which you think conditional binomial probabilities can be applied. (d) Hint for solution: Review item 6, conditional probability, in the summary of basic probability rules at the end of Section \(4.2 .\) Note that $$ P(A \mid B)=\frac{P(A \text { and } B)}{P(B)} $$ and show that in part (a) $$ P(6 \leq r \mid 4 \leq r)=\frac{P((6 \leq r) \text { and }(4 \leq r))}{P(4 \leq r)}=\frac{P(6 \leq r)}{P(4 \leq r)} $$

Step by step solution

01

Define the binomial probabilities for part (a)

For part (a), we have a binomial distribution where the probability of a good crop in one year is \( p = 0.65 \), and the number of trials \( n = 8 \). We want to find the probability of getting at least 6 good years out of 8, given at least 4 good years out of 8: \( P(6 \leq r \mid 4 \leq r) \).

02

Use the conditional probability formula

Recall that the conditional probability is given by \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \). Here, \( A \) is the event \( 6 \leq r \leq 8 \) and \( 4 \leq r \leq 8 \). Hence, \( P(6 \leq r \mid 4 \leq r) = \frac{P(6 \leq r)}{P(4 \leq r)} \).

03

Calculate \(P(6 \leq r)\) for part (a)

To find \( P(6 \leq r) \), calculate \( P(r=6) + P(r=7) + P(r=8) \) for \( r \) following a binomial distribution: \[ P(r=k) = \binom{8}{k} \times 0.65^k \times (1-0.65)^{8-k} \]Compute this for each \( k = 6, 7, 8 \).

04

Compute \(P(4 \leq r)\) for part (a)

Calculate \( P(4 \leq r) \) by summing probabilities for \( r=4, 5, 6, 7, 8 \) using the same binomial probability formula and add the results.

05

Compute the probability for part (b)

For part (b) with \( n = 10 \), \( p = 0.65 \), compute \( P(8 \leq r \mid 6 \leq r) \) using the conditional probability formula: \[ P(8 \leq r \mid 6 \leq r) = \frac{P(8 \leq r)}{P(6 \leq r)} \].Calculate \( P(8 \leq r) \) and \( P(6 \leq r) \) with the binomial distribution, using similar steps from Step 3 and Step 4 but for \( n = 10 \).

06

Apply binomial distributions to other areas in part (c)

Conditional binomial probabilities can apply in various fields. Examples include: 1. Quality control in manufacturing: forecasting defective products in batches given a probability of defects. 2. Marketing: predicting customer purchases over time given initial purchase thresholds. 3. Health care: assessing patient outcomes based on the probability of previous similar cases diagnosed correctly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution

The binomial distribution is a statistical method used to model the number of successes in a fixed number of independent trials, with a constant probability of success in each trial. It's particularly useful in scenarios where we want to predict outcomes that are dichotomous, meaning they have only two possible results, such as success or failure.
For instance, in agriculture, the binomial distribution can model the probability of having a good crop season. Each year can be seen as a trial with a probability of success (a good crop) based on historical data. For the Zimmer farm, each year of having a good crop is a success, with the probability of success being 0.65, which is typical for regions relying on rain and snow.
Calculating binomial probabilities involves using the formula:

• \( P(k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \)

Where \( \binom{n}{k} \) is the combination formula, \( n \) is the number of trials (years), \( k \) is the number of successful outcomes (good crop years), and \( p \) is the probability of success in each trial.
Understanding this distribution helps farmers, like those at Zimmer, project and plan for successful crop yields.

Agricultural Statistics

Agricultural statistics play a vital role in planning and decision-making in farming. By collecting and analyzing data on weather patterns, soil quality, water availability, and historical crop yields, farmers can make informed predictions about future crops.
In regions such as the western United States, agricultural statistics are crucial due to the reliance on winter snow and spring rain for successful wheat production. These statistics help farmers like the Zimmer and Montoya families understand the probability of having good crop years and prepare their strategies and investments accordingly.
Utilizing agricultural statistics allows farmers to mitigate risk, improve financial planning, and optimize resource use. By understanding the likelihood of different agricultural outcomes, they can better predict their financial security and sustainability over years with varying climatic conditions.

Conditional Binomial Probabilities

Conditional binomial probabilities are a subsector of probability dealing with the likelihood of an event happening, given that another event has already occurred. This concept is particularly relevant in situations where decisions are influenced by prior outcomes.
For example, in the question of whether a farm will have a certain number of good years given previous good years, conditional probabilities allow for more accurate forecasting. In our example, the probability that the Zimmer farm has at least six good years, given they already expect at least four good years, is crucial for deciding on financial and operational strategies.
Using conditional probability involves applying the formula:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)

Where \( P(A \mid B) \) is the probability of event A given event B has occurred. This allows for precise calculations that not only help with risk assessment but also with making evidence-based predictions.

Probability Rules

Probability rules form the foundation of probabilistic analysis and include a variety of principles such as the addition rule, multiplication rule, and conditional probability rule. These rules are essential for evaluating the likelihood of events occurring in combination or in sequence.
The conditional probability rule is particularly useful in scenarios like farming, where the success of a future crop season may depend on past conditions or expectations. This is evident in the exercises where both the Zimmer and Montoya farms rely on conditional probability to gauge their likelihood of having enough good crop years for financial viability.
Understanding and correctly applying these rules enable farmers to make informed decisions, taking into account all possible outcomes and their interdependencies. For instance, by knowing that conditional probabilities depend on dividing the probability of two events happening together by the probability of the known event, farmers can tailor their strategies and ensure better preparedness for future agricultural seasons.