Question

Seeds are often treated with a fungicide for protection in poor-draining, wet environments. In a small-scale trial prior to a large-scale experiment to determine what dilution of the fungicide to apply, five treated seeds and five untreated seeds were planted in clay soil and the number of plants emerging from the treated and untreated seeds were recorded. Suppose the dilution was not effective and only four plants emerged. Let \(x\) represent the number of plants that emerged from treated seeds. a. Find the probability that \(x=4\). b. Find \(P(x \leq 3)\). c. Find \(P(2 \leq x \leq 3)\).

Short answer

a. The probability that exactly 4 plants emerged from treated seeds is approximately 0.4096. b. The probability that at most 3 plants emerged from treated seeds is approximately 0.5142. c. The probability that the number of emerged plants from treated seeds is between 2 and 3 inclusive is approximately 0.4934.

Step-by-step solution

Identify the Constants

In this problem, we know that there were five treated seeds, so the number of trials \(n = 5\). We also know that only four plants were emerged, so we need to find the probability of success \(p\). Since the dilution was not effective, we assume the probability of success for each trial is equal, that is: \(p = \frac{4}{5}\) Now we have all the constants we need to use the binomial probability formula.

Find the Probability x=4

Using the binomial probability formula, we can find the probability that exactly four plants emerged from treated seeds: \(P(x=4) = \binom{5}{4} (\frac{4}{5})^4(1-\frac{4}{5})^{5-4}\) \(P(x=4) = \binom{5}{4} (\frac{4}{5})^4(\frac{1}{5})^1\) Calculate the binomial coefficient: \(\binom{5}{4} = \frac{5!}{4!(5-4)!} = 5\) Now compute the probability: \(P(x=4) = 5(\frac{4}{5})^4(\frac{1}{5})^1 \approx 0.4096\)

Find the Probability x≤3

In this part, we are asked to find the probability that the number of emerged plants from treated seeds is less than or equal to 3: \(P(x \leq 3) = P(x=0) + P(x=1) + P(x=2) + P(x=3)\) Calculate each probability using the binomial probability formula: \(P(x=0) = \binom{5}{0} (\frac{4}{5})^0(\frac{1}{5})^5\) \(P(x=1) = \binom{5}{1} (\frac{4}{5})^1(\frac{1}{5})^4\) \(P(x=2) = \binom{5}{2} (\frac{4}{5})^2(\frac{1}{5})^3\) \(P(x=3) = \binom{5}{3} (\frac{4}{5})^3(\frac{1}{5})^2\) Compute the probabilities and sum them up: \(P(x \leq 3) = 0.00032 + 0.02048 + 0.16448 + 0.32896 \approx 0.5142\)

Find the Probability 2≤x≤3

For this part, we need to find the probability that the number of emerged plants from treated seeds is between 2 and 3 inclusive: \(P(2 \leq x \leq 3) = P(x=2) + P(x=3)\) We already found these probabilities in the previous step, so we can simply add them up: \(P(2 \leq x \leq 3) = 0.16448 + 0.32896 \approx 0.4934\) #Final Results# a. The probability that \(x=4\) is approximately \(0.4096\) b. The probability that \(x\leq 3\) is approximately \(0.5142\) c. The probability that \(2\leq x \leq 3\) is approximately \(0.4934\)

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random events. It provides a framework to predict and quantify the likelihood of various outcomes. In simple terms, probability helps us understand how often an event will occur in the long run, if repeated many times. In our seed planting example, probability theory allows us to judge how effective the fungicide treatment might be, given our knowledge of emerging plants.

Here are some key points about probability theory:

• Probability ranges from 0 to 1, where 0 means the event cannot occur, and 1 means the event is certain to happen.
• The sum of probabilities of all potential outcomes of a random event equals 1.
• Probability can be calculated for a single event or a series of events, considering how events influence each other.

Considering the seeds example, probability theory helps us calculate the chance that a specific number of plants (0 to 5) emerge from treated seeds.

Binomial Coefficient

The binomial coefficient, denoted as \(\binom{n}{k}\), is a fundamental concept in combinatorics, the field of mathematics concerned with counting and arrangement. It calculates the number of ways \(k\) successes can occur in \(n\) trials. This is crucial in problems involving binomial distributions, such as our seed experiment.

To understand how it works, consider the process of calculating \(\binom{5}{4}\) as seen in the step-by-step solution. This coefficient is computed through the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, \(n!\) ("n factorial") means \(n\) multiplied by every positive integer smaller than itself, stopping at 1.

When applying the coefficient \(\binom{5}{4}\), it tells us there are 5 possible ways to choose 4 emerging plants out of 5 trials. These ways consider all possible arrangements for successes and failures.

Probability Distribution

A probability distribution maps out the probabilities of all possible outcomes of a random variable. It's a comprehensive picture showing how likely different outcomes are. In the context of our seeds exercise, the binomial distribution is used because each trial has only two possible outcomes: success (a plant emerges) or failure (no plant emerges).

The binomial distribution formula is:
\[P(x=k) = \binom{n}{k} p^k (1-p)^{n-k}\]

• Here, \(P(x=k)\) indicates the probability of exactly \(k\) successes (plants emerging) out of \(n\) trials (seed plantings).
• \(p\) is the probability of success on a single trial, and \((1-p)\) the probability of failure.

Probability distributions, like the binomial, not only help in calculating specific probabilities (as seen in the exercise) but also aid in understanding the behavior and variability of random variables over repeated trials.

Random Variable

A random variable is a variable whose value is subject to variations due to randomness. Each outcome of a random variable is assigned a specific probability. In our seeds exercise, the random variable is represented by \(x\), the number of plants emerging from treated seeds.

Here's how it works:

• We define \(x\) to count different outcomes, such as how many plants will emerge from the 5 treated seeds.
• The random variable \(x\) can take on values ranging from 0 to 5, corresponding to possible success results.
• Each value that \(x\) can assume has a probability associated with it, determined using the probability distribution.

Random variables are essential tools in probability theory because they connect theoretical probabilities with practical experiments, like seed germination rate studies. Through understanding random variables, we can assess and predict outcomes in uncertain situations effectively.