Problem 21 - 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 four seeds in each hill. (a) What is the probability that at least one seed will germinate in a given hill? (b) What is the probability that two or more seeds will germinate in a given hill? (c) What is the probability that all four seeds will germinate in a given hill?

Step-by-Step Solution

Answer

(a) 0.9961; (b) 0.9492; (c) 0.3164.

Step 1: Probability of No Seeds Germinating

To find the probability of at least one seed germinating, first calculate the probability of none of the seeds germinating. Since the probability of a seed not germinating is 1 - 0.75 = 0.25, the probability that all four seeds do not germinate is \((0.25)^4 = 0.00390625\).

Step 2: Probability of At Least One Seed Germinating

To find the probability of at least one seed germinating, subtract the probability of no seeds germinating from 1. Thus, the probability is \(1 - 0.00390625 = 0.99609375\).

Step 3: Probability of Two or More Seeds Germinating

Calculate the complementary probability that less than two seeds (either 0 or 1 seed) germinate. We've already calculated the probability of no seeds germinating, which is \(0.00390625\). For only one seed germinating, the probability is calculated using the binomial formula: \(\binom{4}{1} (0.75)^1 (0.25)^3 = 0.046875\). Adding these gives the probability of less than two seeds germinating: \(0.00390625 + 0.046875 = 0.05078125\). Therefore, the probability that two or more seeds germinate is \(1 - 0.05078125 = 0.94921875\).

Step 4: Probability of All Seeds Germinating

The probability that all four seeds will germinate is simply \((0.75)^4 = 0.31640625\).

Key Concepts

Understanding Binomial Probability

Binomial probability is a fundamental concept in probability theory. It deals with situations where only two outcomes are possible, like success and failure. These are often referred to as 'yes' or 'no' answers. For instance, in our seed germination example: a seed either germinates (success) or it does not (failure).
Binomial probability is used to calculate the likelihood of achieving a fixed number of successes in a specified number of trials. Each trial should be independent, meaning the outcome of one does not affect the others.
The binomial formula is pivotal for these calculations. It is: \[ P(X = k) = \binom{n}{k} (p)^k (1 - p)^{n-k} \] where:

• \( n \) is the number of trials,
• \( k \) is the number of successful outcomes,
• \( p \) is the probability of success on an individual trial, and
• \( 1-p \) represents the probability of failure.

This formula helps us determine the probability of different scenarios in our trials, such as the probability of exactly two or more seeds germinating.

Exploring Germination Rates

Germination rate refers to the likelihood that a seed will sprout and grow. It's usually expressed as a decimal or percentage. For example, a germination rate of 0.75 indicates a 75% chance that any individual seed will germinate.
Understanding germination rates is vital for gardeners and agricultural scientists because it helps in predicting outcomes with greater accuracy. In our case with tomato seeds, with a high germination rate like 0.75, we can expect most seeds to grow successfully.
By planting multiple seeds in a single hill, gardeners increase their chances of achieving at least one successful plant. This method leverages probability to minimize the risk of having no plants at all in a hill, sparked by the reliability of the high germination rate.

Complementary Probability Simplified

Complementary probability is a simple yet powerful concept. It involves calculating the probability of events by considering their complementary (opposite) events. The main idea is that the total probability of both an event and its complement is always 1.
For instance, if we want the probability of at least one seed germinating, we first find the probability that none germinate. In our tomato seed example, the probability of no seeds germinating is 0.00390625, calculated as \((0.25)^4\). Then, the complementary probability of at least one seed germinating is \(1 - 0.00390625 = 0.99609375\).
This method allows for easier calculations of complex probabilities, as sometimes calculating the complement is simpler than directly calculating the desired probability. This strategic approach is a cornerstone in probability problems.