Problem 19 - Algebra and Trigonometry

Question

Germination Rates \(\quad\) A certain brand of tomato seeds has a 0.75 probability of germinating. To increase the chance that at least one tomato plant per seed hill germinates, a gardener plants 4 seeds in each hill. (a) What is the probability that least one seed germinates in a given hill? (b) What is the probability that 2 or more seeds will germinate in a given hill? (c) What is the probability that all 4 seeds germinate in a given hill?

Step-by-Step Solution

Answer

(a) 0.996, (b) 0.949, (c) 0.316

Step 1: Calculate Probability of No Seed Germinating

The probability that a single seed does not germinate is \[ q = 1 - 0.75 = 0.25 \] Thus, the probability that all 4 seeds do not germinate is \[ P(\text{no seeds germinate}) = 0.25^4 \] Calculating this gives: \[ P(\text{no seeds germinate}) = 0.00390625 \]

Step 2: Calculate Probability of At Least One Seed Germinating

The probability that at least one seed germinates is the complement of the probability that no seeds germinate: \[ P(\text{at least one seed germinates}) = 1 - P(\text{no seeds germinate}) = 1 - 0.00390625 = 0.99609375 \]

Step 3: Calculate Probability of Exactly One Seed Germinating

The probability that exactly one seed germinates can be calculated using the binomial distribution formula: \[ P(X=1) = \binom{4}{1} \cdot (0.75)^1 \cdot (0.25)^3 = 4 \cdot 0.75 \cdot 0.015625 = 0.046875 \]

Step 4: Calculate Probability of 2 or More Seeds Germinating

The probability that 2 or more seeds germinate is \[ P(2 \text{ or more germinate}) = 1 - P(\text{0 seeds germinate}) - P(1 \text{ seed germinates}) = 1 - 0.00390625 - 0.046875 = 0.94921875 \]

Step 5: Calculate Probability of All 4 Seeds Germinating

The probability that all 4 seeds germinate is \[ P(\text{all 4 seeds germinate}) = (0.75)^4 = 0.31640625 \]

Key Concepts

Germination Rate

Germination rate refers to the likelihood of a seed successfully sprouting and beginning to grow. For the tomato seeds discussed in the exercise, the probability of germination is 0.75, or 75%. This is a quite favorable rate, indicating that if you plant 100 seeds, you can expect about 75 to germinate on average.

Understanding germination rates helps gardeners and farmers decide how densely to plant seeds. A higher germination rate means fewer seeds are needed to ensure a full crop, but it also highlights the importance of planting multiple seeds in places where at least one plant needs to grow.

• Significance of Germination Rate: A better rate increases the chance of plant success.
• Practical Application: Knowing the rate allows planning for adequate planting density.
• Probability in Germination: Expresses in percentages or decimal figures like 0.75 (75%).

Binomial Distribution

The binomial distribution is a statistical method used to model the number of successful outcomes, such as seed germinations, in a set number of trials. In the exercise, there are 4 seeds planted in each hill, which serves as the number of trials. An outcome like a seed germinating is treated as a 'success'.

This distribution is especially useful in this case because it calculates the probabilities of different numbers of successes (germinations) with known probabilities. For example, if the goal is at least one seed germinating, the binomial distribution is used to compute the likelihood of 0, 1, 2, 3, or 4 seeds germinating.

• Formula: The binomial probability formula is \[ P(X=k) = \binom{n}{k} \, p^k \, (1-p)^{n-k} \]
• Variables: Here, \( n = 4 \) (number of seeds), \( k \) (number of successes), \( p = 0.75 \) (probability of one success).
• Uses: Providing probabilities for different numbers of germinated seeds.

Complementary Probability

Complementary probability is a straightforward concept in probability theory where the probability of an event not happening is the complement of it happening. It utilizes the idea that the total probability of all possible outcomes must equal 1. In the exercise, rather than calculating directly the probability of at least one seed germinating, complementary probability is used.

By first finding the probability that no seeds germinate and subtracting it from 1, we easily find the probability of the complementary event — at least one seed germinating.

• Formula: The complement rule is \[ P(A^c) = 1 - P(A) \]
• Role in Solution: Easier calculation of at least one seed germinating.
• Simplifies Complex Problems: Often faster than calculating the direct outcome.

Exact Probability Calculation

Exact probability calculations allow us to find the precise likelihood of specific outcomes. For example, when determining the probability of exactly one seed germinating, the binomial distribution formula is used based on exact specifications. This calculation is crucial in comprehending the detailed likelihood of each specific outcome in a trial.

Calculations can serve different needs, such as determining the probability of exactly two or more seeds germinating or all seeds germinating as found in several steps of the solution. Exact probabilities provide a clear view on what to expect under particular conditions, aiding in effective planning.

• Specificity: Provides precise probabilities like exactly one or all four seeds germinating.
• Methodical Approach: Employs the binomial formula to ensure accuracy.
• Application: Useful for planning seed planting strategies for maximum yield.